Extremum-seeking control system for a plant

ABSTRACT

An extremum-seeking control system for a plant includes a feedback controller for operating the plant to achieve a value of a manipulated variable, and an extremum-seeking controller. The extremum-seeking controller is configured to provide the value of the manipulated variable to the feedback controller and to determine a value for the manipulated variable. The extremum-seeking controller determines the value for the manipulated variable by perturbing the manipulated variable with an excitation signal and monitoring a performance variable of the plant resulting from the perturbed manipulated variable, estimating a normalized correlation coefficient relating the performance variable to the manipulated variable, and modulating the manipulated variable to drive the normalized correlation coefficient toward zero using a general set of tuning parameters. The general set of tuning parameters are adapted for use with the normalized correlation coefficient, independent of a scale of the performance variable.

CROSS-REFERENCE TO RELATED PATENT APPLICATION

This application is a continuation of U.S. patent application Ser. No.15/080,435 filed Mar. 24, 2016, which claims the benefit of and priorityto U.S. Provisional Patent Application No. 62/296,713 filed Feb. 18,2016, the entire disclosures of which are incorporated by referenceherein.

BACKGROUND

The present disclosure relates generally to an extremum-seeking control(ESC) system. ESC is a class of self-optimizing control strategies thatcan dynamically search for the unknown and/or time-varying inputs of asystem for optimizing a certain performance index. ESC can be considereda dynamic realization of gradient searching through the use of dithersignals. The gradient of the system output y with respect to the systeminput u can be obtained by slightly perturbing the system operation andapplying a demodulation measure. Optimization of system performance canbe obtained by driving the gradient toward zero by using a negativefeedback loop in the closed-loop system. ESC is a non-model basedcontrol strategy, meaning that a model for the controlled system is notnecessary for ESC to optimize the system.

A plant in control theory is the combination of a process and one ormore mechanically-controlled outputs. Traditional ESC systems typicallyuse a periodic (e.g., sinusoidal) dither signal to perturb a controlinput u provided to the plant. The periodic dither signal can causelarge variations (e.g., sinusoidal oscillations) in both the controlinput u and the plant output y, which are often noticeable to plantoperators. Additionally, the frequency of a periodic dither signal needsto be carefully selected to ensure that the ESC strategy is effective.For example, it may be desirable to select a dither signal frequencybased on the natural frequency of the plant to enhance the effect of thedither signal on the plant output y. It can be difficult and challengingto properly select the dither frequency without knowledge of thedynamics of the plant. For these reasons, the use of a periodic dithersignal is one of the drawbacks of traditional ESC.

SUMMARY

One implementation of the present disclosure is a control system for aplant. The control system include a feedback controller and anextremum-seeking controller, according to some embodiments. The feedbackcontroller is for operating the plant to achieve a value of amanipulated variable, according to some embodiments. Theextremum-seeking controller is configured to provide the value of themanipulated variable to the feedback controller and to determine a valuefor the manipulated variable, according to some embodiments. In someembodiments, the extremum-seeking controller determines the value forthe manipulated variable by perturbing the manipulated variable with anexcitation signal and monitoring a performance variable of the plantresulting from the perturbed manipulated variable. In some embodiments,the extremum-seeking controller determines the value for the manipulatedvariable by estimating a normalized correlation coefficient relating theperformance variable to the manipulated variable, and modulating themanipulated variable to drive the normalized correlation coefficienttoward zero using a general set of tuning parameters. In someembodiments, the general set of tuning parameters are adapted for usewith the normalized correlation coefficient, independent of a scale ofthe performance variable.

In some embodiments, the excitation signal is a non periodic signalincluding at least one of a random walk signal, a non deterministicsignal, and a non-repeating signal.

In some embodiments, the value of the manipulated variable includes astochastic portion defined by a stochastic excitation signal, and anon-stochastic portion determined by driving the normalized correlationcoefficient to zero.

In some embodiments, the extremum-seeking controller includes anintegrator configured to generate the excitation signal by integrating arandom noise signal.

In some embodiments, the extremum-seeking controller is configured toestimate the normalized correlation coefficient relating the performancevariable to the manipulated variable by performing a recursive leastsquares estimation process with exponential forgetting.

In some embodiments, the extremum-seeking controller is configured toestimate a gradient of the performance variable with respect to themanipulated variable by performing a regression process.

In some embodiments, the regression process includes obtaining a linearmodel for the performance variable, the linear model defining theperformance variable as a linear function of the manipulated variable,an offset parameter, and a gradient parameter. In some embodiments, theregression process also includes estimating a value for the gradientparameter based on an observed value for the performance variable and anobserved value for the manipulated variable. In some embodiments, theregression progress also includes using the estimated value for thegradient parameter as the gradient of the performance variable withrespect to the manipulated variable.

In some embodiments, the feedback controller is configured to achievethe manipulated variable by adjusting operation of equipment of theplant.

Another implementation of the present disclosure is an extremum-seekingcontroller for a plant. In some embodiments, the extremum-seekingcontroller includes one or more processors and one or morenon-transitory computer-readable media storing instructions that, whenexecuted by the one or more processors, cause the one or more processorsto perform operations. In some embodiments, the operations includeperturbing a manipulated variable with an excitation signal andproviding the perturbed manipulated variable as an input to a plant. Insome embodiments, the operations also include monitoring a performancevariable of the plant resulting from the perturbed manipulated variable.In some embodiments, the operations also include estimating a normalizedcorrelation coefficient relating the performance variable to themanipulated variable. In some embodiments, the operations also includemodulating the manipulated variable to drive the normalized correlationcoefficient toward zero using a general set of tuning parameters. Insome embodiments, the general set of tuning parameters are adapted foruse with the normalized correlation coefficient, independent of a scaleof the performance variable.

In some embodiments, the normalized correlation coefficient is estimatedwith a recursive estimation process. In some embodiments, the recursiveestimation process is a recursive least squares estimation process withexponential forgetting.

In some embodiments, the one or more processors are configured toperform the recursive estimation process for the manipulated variable bycalculating a covariance between the performance variable and themanipulated variable, calculating a variance of the manipulatedvariable, and using the calculated covariance and the calculatedvariance to estimate the normalized correlation coefficient relating theperformance variable to the manipulated variable.

In some embodiments, the one or more processors are configured toperform the recursive estimation process for the manipulated variable bycalculating an exponentially-weighted moving average (EWMA) of aplurality of samples of the manipulated variable. In some embodiments,the one or more processors are configured to perform the recursiveestimation process for the manipulated variable by calculating an EWMAof a plurality of samples of the performance variable, and using theEWMAs to estimate the normalized correlation coefficient relating theperformance variable to the manipulated variable.

In some embodiments, the recursive estimation process is a regressionprocess.

In some embodiments, the one or more processors are configured toperform the regression process by obtaining a linear model for theperformance variable, the linear model defining the performance variableas a linear function of the manipulated variable and a gradientparameter for the manipulated variable. In some embodiments, the one ormore processors are configured to perform the regression process byestimating a value for the gradient parameter based on observed valuefor the manipulated variable and an observed value for the performancevariable, and using the estimated values for the gradient parameter asthe normalized correlation coefficients relating the performancevariable to the manipulated variable.

In some embodiments, the excitation signal is a non periodic signalincluding at least one of a random walk signal, a non deterministicsignal, and a non-repeating signal.

Another implementation of the present disclosure is an extremum-seekingcontroller for a plant. In some embodiments, the extremum-seekingcontroller includes one or more processors, and one or morenon-transitory computer-readable media storing instructions that, whenexecuted by the one or more processors, cause the one or more processorsto perform operations. In some embodiments, the operations includeperturbing each of a plurality of manipulated variables with a differentexcitation signal. In some embodiments, the operations includemonitoring a performance variable of the plant resulting from each ofthe perturbed manipulated variables. In some embodiments, the operationsinclude estimating normalized correlation coefficients relating theperformance variable to each of the perturbed manipulated variables, andmodulating the manipulated variables to drive the estimated normalizedcorrelation coefficients toward zero using a general set of tuningparameters. In some embodiments, the general set of tuning parametersare adapted for use with the normalized correlation coefficient,independent of a scale of the performance variable.

In some embodiments, the one or more processors are configured toestimate the normalized correlation coefficient for each manipulatedvariable by calculating a covariance between each of the manipulatedvariables and the performance variable. In some embodiments, the one ormore processors are configured to estimate the normalized correlationcoefficient for each manipulated variable by calculating a variance ofeach of the manipulated variables, calculating a variance of theperformance variable, and using the calculated covariance and thecalculated variances to estimate the normalized correlation coefficient.

In some embodiments, the one or more processors are configured toestimate the normalized correlation coefficient for each of themanipulated variables by estimating a gradient of the performancevariable with respect to each of the manipulated variables. In someembodiments, the one or more processors are configured to estimate thenormalized correlation coefficient foe ach of the manipulated variablesby calculating a standard deviation of each of the manipulatedvariables, calculating a standard deviation of the performance variable,and using the estimated gradient and the calculated standard deviationsto estimate the normalized correlation coefficient.

In some embodiments, the one or more processors are configured toestimate the normalized correlation coefficient for each of themanipulated variables by calculating an exponentially-weighted movingaverage (EWMA) of a plurality of samples of each of the manipulatedvariables. In some embodiments, the one or more processors areconfigured to estimate the normalized correlation coefficient for eachof the manipulated variables by calculating an EWMA of a plurality ofsamples of the performance variable, and using the EWMAs to estimate thenormalized correlation coefficient.

In some embodiments, the excitation signal is a non periodic signalincluding at least one of a random walk signal, a non deterministicsignal, and a non-repeating signal.

Those skilled in the art will appreciate that the summary isillustrative only and is not intended to be in any way limiting. Otheraspects, inventive features, and advantages of the devices and/orprocesses described herein, as defined solely by the claims, will becomeapparent in the detailed description set forth herein and taken inconjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a drawing of a building in which an extremum-seeking controlsystem can be implemented, according to some embodiments.

FIG. 2 is a block diagram of a building HVAC system in which anextremum-seeking control system can be implemented, according to someembodiments.

FIG. 3 is a block diagram of an extremum-seeking control system whichuses a periodic dither signal to perturb a control input provided to aplant, according to some embodiments.

FIG. 4 is a block diagram of another extremum-seeking control systemwhich uses a periodic dither signal to perturb a control input providedto a plant, according to some embodiments.

FIG. 5 is a block diagram of an extremum-seeking control system whichuses a stochastic dither signal to perturb a control input provided to aplant and a recursive estimation technique to estimate a gradient orcoefficient relating an output of the plant to the control input,according to some embodiments.

FIG. 6A is a graph of a control input provided to a plant, illustratingperiodic oscillations caused by perturbing the control input with aperiodic dither signal, according to some embodiments.

FIG. 6B is a graph of a performance variable received from the plantresulting from the perturbed control input shown in FIG. 6A, accordingto some embodiments.

FIG. 7A is a graph of a control input provided to a plant when astochastic excitation signal is used to perturb the control input,according to some embodiments.

FIG. 7B is a graph of a performance variable received from the plantresulting from the perturbed control input shown in FIG. 7A, accordingto some embodiments.

FIG. 8 is a flow diagram illustrating an extremum-seeking controltechnique in which a stochastic excitation signal is used to perturb acontrol input to a plant, according to some embodiments.

FIG. 9 is a flow diagram illustrating an extremum-seeking controltechnique in which normalized correlation coefficient is used to relatea performance variable received from the plant to a control inputprovided to the plant, according to some embodiments.

FIG. 10A is a block diagram of a chilled water plant in which thesystems and methods of the present disclosure can be implemented,according to some embodiments.

FIG. 10B is a flow diagram illustrating an extremum-seeking controltechnique in which a stochastic excitation signal is used to perturb acondenser water temperature setpoint in the chilled water plant of FIG.10A, according to some embodiments.

FIG. 10C is a flow diagram illustrating an extremum-seeking controltechnique in which a normalized correlation coefficient is used torelate the total system power consumption to the condenser watertemperature setpoint in the chilled water plant of FIG. 10A, accordingto some embodiments.

FIG. 11A is a block diagram of another chilled water plant in which thesystems and methods of the present disclosure can be implemented,according to some embodiments.

FIG. 11B is a flow diagram illustrating an extremum-seeking controltechnique in which stochastic excitation signals are used to perturbcondenser water pump speed and a cooling tower fan speed in the chilledwater plant of FIG. 11A, according to some embodiments.

FIG. 11C is a flow diagram illustrating an extremum-seeking controltechnique in which normalized correlation coefficients are used torelate the total system power consumption to the condenser water pumpspeed and the cooling tower fan speed in the chilled water plant of FIG.11A, according to some embodiments.

FIG. 12A is a block diagram of a variable refrigerant flow system inwhich the systems and methods of the present disclosure can beimplemented, according to some embodiments.

FIG. 12B is a flow diagram illustrating an extremum-seeking controltechnique in which a stochastic excitation signal is used to perturb arefrigerant pressure setpoint in the variable refrigerant flow system ofFIG. 12A, according to some embodiments.

FIG. 12C is a flow diagram illustrating an extremum-seeking controltechnique in which a normalized correlation coefficient is used torelate the total system power consumption to the refrigerant pressuresetpoint in the variable refrigerant flow system of FIG. 12A, accordingto some embodiments.

FIG. 13A is a block diagram of another variable refrigerant flow systemin which the systems and methods of the present disclosure can beimplemented, according to some embodiments.

FIG. 13B is a flow diagram illustrating an extremum-seeking controltechnique in which stochastic excitation signals are used to arefrigerant pressure setpoint and a refrigerant superheat setpoint inthe variable refrigerant flow system of FIG. 13A, according to someembodiments.

FIG. 13C is a flow diagram illustrating an extremum-seeking controltechnique in which normalized correlation coefficients are used torelate the total system power consumption to the refrigerant pressuresetpoint and the refrigerant superheat setpoint in the variablerefrigerant flow system of FIG. 13A, according to some embodiments.

FIG. 14A is a block diagram of a vapor compression system in which thesystems and methods of the present disclosure can be implemented,according to some embodiments.

FIG. 14B is a flow diagram illustrating an extremum-seeking controltechnique in which a stochastic excitation signal is used to perturb asupply air temperature setpoint in the vapor compression system of FIG.14A, according to some embodiments.

FIG. 14C is a flow diagram illustrating an extremum-seeking controltechnique in which a normalized correlation coefficient is used torelate the total system power consumption to the supply air temperaturesetpoint in the vapor compression system of FIG. 14A, according to someembodiments.

FIG. 15A is a block diagram of another vapor compression system in whichthe systems and methods of the present disclosure can be implemented,according to some embodiments.

FIG. 15B is a flow diagram illustrating an extremum-seeking controltechnique in which a stochastic excitation signal is used to perturb anevaporator fan speed in the vapor compression system of FIG. 15A,according to some embodiments.

FIG. 15C is a flow diagram illustrating an extremum-seeking controltechnique in which a normalized correlation coefficient is used torelate the total system power consumption to the evaporator fan speed inthe vapor compression system of FIG. 15A, according to some embodiments.

FIG. 16A is a block diagram of another vapor compression system in whichthe systems and methods of the present disclosure can be implemented,according to some embodiments.

FIG. 16B is a flow diagram illustrating an extremum-seeking controltechnique in which stochastic excitation signals are used to perturb asupply air temperature setpoint and a condenser fan speed in the vaporcompression system of FIG. 16A, according to some embodiments.

FIG. 16C is a flow diagram illustrating an extremum-seeking controltechnique in which normalized correlation coefficients are used torelate the total system power consumption to the supply air temperaturesetpoint and the condenser fan speed in the vapor compression system ofFIG. 16A, according to some embodiments.

DETAILED DESCRIPTION Overview

Referring generally to the FIGURES, various extremum-seeking control(ESC) systems and methods are shown, according to some embodiments. Ingeneral, ESC is a class of self-optimizing control strategies that candynamically search for the unknown and/or time-varying inputs of asystem for optimizing a certain performance index. ESC can be considereda dynamic realization of gradient searching through the use of dithersignals. The gradient of the system output y with respect to the systeminput u can be obtained by slightly perturbing the system operation andapplying a demodulation measure.

Optimization of system performance can be obtained by driving thegradient toward zero by using a feedback loop in the closed-loop system.ESC is a non-model based control strategy, meaning that a model for thecontrolled system is not necessary for ESC to optimize the system.Various implementations of ESC are described in detail in U.S. Pat. Nos.8,473,080, 7,827,813, 8,027,742, 8,200,345, 8,200,344, U.S. patentapplication Ser. No. 14/495,773, U.S. patent application Ser. No.14/538,700, U.S. patent application Ser. No. 14/975,527, and U.S. patentapplication Ser. No. 14/961,747. Each of these patents and patentapplications is incorporated by reference herein.

In some embodiments, an extremum-seeking controller uses a stochasticexcitation signal q to perturb a control input u provided to a plant.The controller can include a stochastic signal generator configured togenerate a stochastic signal. The stochastic signal can be a randomsignal (e.g., a random walk signal, a white noise signal, etc.), anon-periodic signal, an unpredictable signal, a disturbance signal, orany other type of non-deterministic or non-repeating signal. In someembodiments, the stochastic signal has a non-zero mean. The stochasticsignal can be integrated to generate the excitation signal q.

The stochastic excitation signal q can provide variation in the controlinput u sufficient to estimate the gradient of the plant output (i.e., aperformance variable y) with respect to the control input u. Thestochastic excitation signal q has several advantages over a traditionalperiodic dither signal v. For example, the stochastic excitation signalq is less perceptible than the traditional periodic dither signal v. Assuch, the effects of the stochastic excitation signal q on the controlinput u are less noticeable than the periodic oscillations caused by thetraditional periodic dither signal v. Another advantage of thestochastic excitation signal q is that tuning the controller is simplerbecause the dither frequency ω_(v) is no longer a required parameter.Accordingly, the controller does not need to know or estimate thenatural frequency of the plant when generating the stochastic excitationsignal q.

In some embodiments, the extremum-seeking controller uses a recursiveestimation technique to estimate the gradient of the performancevariable y with respect to the control input u. For example, thecontroller can use a recursive least-squares (RLS) estimation techniqueto generate an estimate of the gradient

$\frac{dy}{du}.$

In some embodiments, the controller uses exponential forgetting as partof the RLS estimation technique. For example, the controller can beconfigured to calculate exponentially-weighted moving averages (EWMAs)of the performance variable y, the control input u, and/or othervariables used in the recursive estimation technique. Exponentialforgetting reduces the required amount of data storage (relative tobatch processing) and allows the controller to remain more sensitive torecent data and thus more responsive to a shifting optimal point.

In some embodiments, the extremum-seeking controller estimates anormalized correlation coefficient ρ relating the performance variable yto the control input u. The correlation coefficient ρ can be related tothe performance gradient

$\frac{dy}{du}\left( {{e.g.},{{proportional}\mspace{14mu} {to}\mspace{14mu} \frac{dy}{du}}} \right)$

but scaled based on the range of the performance variable y. Forexample, the correlation coefficient ρ can be a normalized measure ofthe performance gradient

$\frac{dy}{du}$

scaled to the range −1≤ρ≤1. The normalized correlation coefficient ρ canbe estimated based on the covariance between the performance variable yand the control input u, the variance of the performance variable y, andthe variance of the control input u. In some embodiments, the normalizedcorrelation coefficient ρ can be estimated using a recursive estimationprocess.

The correlation coefficient ρ can be used by the feedback controllerinstead of the performance gradient

$\frac{dy}{du}.$

For example, the feedback controller can adjust the DC value w of thecontrol input u to drive the correlation coefficient ρ to zero. Oneadvantage of using the correlation coefficient ρ in place of theperformance gradient

$\frac{dy}{du}$

is that the tuning parameters used by the feedback controller can be ageneral set of tuning parameters which do not need to be customized oradjusted based on the scale of the performance variable y. Thisadvantage eliminates the need to perform control-loop-specific tuningfor the feedback controller and allows the feedback controller to use ageneral set of tuning parameters that are applicable across manydifferent control loops and/or plants. Additional features andadvantages of the extremum-seeking controller are described in greaterdetail below.

Building and HVAC System

Referring now to FIGS. 1-2, a building 10 and HVAC system 20 in which anextremum-seeking control system can be implemented are shown, accordingto some embodiments. Although the ESC systems and methods of the presentdisclosure are described primarily in the context of a building HVACsystem, it should be understood that ESC is generally applicable to anytype of control system that optimizes or regulates a variable ofinterest. For example, the ESC systems and methods of the presentdisclosure can be used to optimize an amount of energy produced byvarious types of energy producing systems or devices (e.g., powerplants, steam or wind turbines, solar panels, combustion systems, etc.)and/or to optimize an amount of energy consumed by various types ofenergy consuming systems or devices (e.g., electronic circuitry,mechanical equipment, aerospace and land-based vehicles, buildingequipment, HVAC devices, refrigeration systems, etc.).

In various implementations, ESC can be used in any type of controllerthat functions to achieve a setpoint for a variable of interest (e.g.,by minimizing a difference between a measured or calculated input and asetpoint) and/or optimize a variable of interest (e.g., maximize orminimize an output variable). It is contemplated that ESC can be readilyimplemented in various types of controllers (e.g., motor controllers,power controllers, fluid controllers, HVAC controllers, lightingcontrollers, chemical controllers, process controllers, etc.) andvarious types of control systems (e.g., closed-loop control systems,open-loop control systems, feedback control systems, feed-forwardcontrol systems, etc.). All such implementations should be consideredwithin the scope of the present disclosure.

Referring particularly to FIG. 1, a perspective view of building 10 isshown. Building 10 is served by HVAC system 20. HVAC system 20 is shownto include a chiller 22, a boiler 24, a rooftop cooling unit 26, and aplurality of air-handling units (AHUs) 36. HVAC system 20 uses a fluidcirculation system to provide heating and/or cooling for building 10.The circulated fluid can be cooled in chiller 22 or heated in boiler 24,depending on whether cooling or heating is required. Boiler 24 may addheat to the circulated fluid by burning a combustible material (e.g.,natural gas). Chiller 22 may place the circulated fluid in a heatexchange relationship with another fluid (e.g., a refrigerant) in a heatexchanger (e.g., an evaporator). The refrigerant removes heat from thecirculated fluid during an evaporation process, thereby cooling thecirculated fluid.

The circulated fluid from chiller 22 or boiler 24 can be transported toAHUs 36 via piping 32. AHUs 36 may place the circulated fluid in a heatexchange relationship with an airflow passing through AHUs 36. Forexample, the airflow can be passed over piping in fan coil units orother air conditioning terminal units through which the circulated fluidflows. AHUs 36 may transfer heat between the airflow and the circulatedfluid to provide heating or cooling for the airflow. The heated orcooled air can be delivered to building 10 via an air distributionsystem including air supply ducts 38 and may return to AHUs 36 via airreturn ducts 40. In FIG. 1, HVAC system 20 is shown to include aseparate AHU 36 on each floor of building 10. In other embodiments, asingle AHU (e.g., a rooftop AHU) may supply air for multiple floors orzones. The circulated fluid from AHUs 36 may return to chiller 22 orboiler 24 via piping 34.

In some embodiments, the refrigerant in chiller 22 is vaporized uponabsorbing heat from the circulated fluid. The vapor refrigerant can beprovided to a compressor within chiller 22 where the temperature andpressure of the refrigerant are increased (e.g., using a rotatingimpeller, a screw compressor, a scroll compressor, a reciprocatingcompressor, a centrifugal compressor, etc.). The compressed refrigerantcan be discharged into a condenser within chiller 22. In someembodiments, water (or another chilled fluid) flows through tubes in thecondenser of chiller 22 to absorb heat from the refrigerant vapor,thereby causing the refrigerant to condense. The water flowing throughtubes in the condenser can be pumped from chiller 22 to a rooftopcooling unit 26 via piping 28. Cooling unit 26 may use fan drivencooling or fan driven evaporation to remove heat from the water. Thecooled water in rooftop unit 26 can be delivered back to chiller 22 viapiping 30 and the cycle repeats.

Referring now to FIG. 2, a block diagram illustrating a portion of HVACsystem 20 in greater detail is shown, according to some embodiments. InFIG. 2, AHU 36 is shown as an economizer type air handling unit.Economizer type air handling units vary the amount of outside air andreturn air used by the air handling unit for heating or cooling. Forexample, AHU 36 may receive return air 82 from building 10 via returnair duct 40 and may deliver supply air 86 to building 10 via supply airduct 38. AHU 36 can be configured to operate exhaust air damper 60,mixing damper 62, and outside air damper 64 to control an amount ofoutside air 80 and return air 82 that combine to form supply air 86. Anyreturn air 82 that does not pass through mixing damper 62 can beexhausted from AHU 36 through exhaust damper 60 as exhaust air 84.

Each of dampers 60-64 can be operated by an actuator. As shown in FIG.2, exhaust air damper 60 is operated by actuator 54, mixing damper 62 isoperated by actuator 56, and outside air damper 64 is operated byactuator 58. Actuators 54-58 may communicate with an AHU controller 44via a communications link 52. AHU controller 44 can be an economizercontroller configured to use one or more control algorithms (e.g.,state-based algorithms, ESC algorithms, PID control algorithms, modelpredictive control algorithms, etc.) to control actuators 54-58.Examples of ESC methods that can be used by AHU controller 44 aredescribed in greater detail with reference to FIGS. 8-9.

Actuators 54-58 may receive control signals from AHU controller 44 andmay provide feedback signals to AHU controller 44. Feedback signals mayinclude, for example, an indication of a current actuator or damperposition, an amount of torque or force exerted by the actuator,diagnostic information (e.g., results of diagnostic tests performed byactuators 54-58), status information, commissioning information,configuration settings, calibration data, and/or other types ofinformation or data that can be collected, stored, or used by actuators54-58.

Still referring to FIG. 2, AHU 36 is shown to include a cooling coil 68,a heating coil 70, and a fan 66. In some embodiments, cooling coil 68,heating coil 70, and fan 66 are positioned within supply air duct 38.Fan 66 can be configured to force supply air 86 through cooling coil 68and/or heating coil 70. AHU controller 44 may communicate with fan 66via communications link 78 to control a flow rate of supply air 86.Cooling coil 68 may receive a chilled fluid from chiller 22 via piping32 and may return the chilled fluid to chiller 22 via piping 34. Valve92 can be positioned along piping 32 or piping 34 to control an amountof the chilled fluid provided to cooling coil 68. Heating coil 70 mayreceive a heated fluid from boiler 24 via piping 32 and may return theheated fluid to boiler 24 via piping 34. Valve 94 can be positionedalong piping 32 or piping 34 to control an amount of the heated fluidprovided to heating coil 70.

Each of valves 92-94 can be controlled by an actuator. As shown in FIG.2, valve 92 is controlled by actuator 88 and valve 94 is controlled byactuator 90. Actuators 88-90 may communicate with AHU controller 44 viacommunications links 96-98. Actuators 88-90 may receive control signalsfrom AHU controller 44 and may provide feedback signals to controller44. In some embodiments, AHU controller 44 receives a measurement of thesupply air temperature from a temperature sensor 72 positioned in supplyair duct 38 (e.g., downstream of cooling coil 68 and heating coil 70).However, temperature sensor 72 is not required and may not be includedin some embodiments.

AHU controller 44 may operate valves 92-94 via actuators 88-90 tomodulate an amount of heating or cooling provided to supply air 86(e.g., to achieve a setpoint temperature for supply air 86 or tomaintain the temperature of supply air 86 within a setpoint temperaturerange). The positions of valves 92-94 affect the amount of cooling orheating provided to supply air 86 by cooling coil 68 or heating coil 70and may correlate with the amount of energy consumed to achieve adesired supply air temperature. In various embodiments, valves 92-94 canbe operated by AHU controller 44 or a separate controller for HVACsystem 20.

AHU controller 44 may monitor the positions of valves 92-94 viacommunications links 96-98. AHU controller 44 may use the positions ofvalves 92-94 as the variable to be optimized using an ESC controltechnique. AHU controller 44 may determine and/or set the positions ofdampers 60-64 to achieve an optimal or target position for valves 92-94.The optimal or target position for valves 92-94 can be the position thatcorresponds to the minimum amount of mechanical heating or cooling usedby HVAC system 20 to achieve a setpoint supply air temperature (e.g.,minimum fluid flow through valves 92-94).

Still referring to FIG. 2, HVAC system 20 is shown to include asupervisory controller 42 and a client device 46. Supervisory controller42 may include one or more computer systems (e.g., servers, BAScontrollers, etc.) that serve as enterprise level controllers,application or data servers, head nodes, master controllers, or fieldcontrollers for HVAC system 20. Supervisory controller 42 maycommunicate with multiple downstream building systems or subsystems(e.g., an HVAC system, a security system, etc.) via a communicationslink 50 according to like or disparate protocols (e.g., LON, BACnet,etc.).

In some embodiments, AHU controller 44 receives information (e.g.,commands, setpoints, operating boundaries, etc.) from supervisorycontroller 42. For example, supervisory controller 42 may provide AHUcontroller 44 with a high fan speed limit and a low fan speed limit. Alow limit may avoid frequent component and power taxing fan start-upswhile a high limit may avoid operation near the mechanical or thermallimits of the fan system. In various embodiments, AHU controller 44 andsupervisory controller 42 can be separate (as shown in FIG. 2) orintegrated. In an integrated implementation, AHU controller 44 can be asoftware module configured for execution by a processor of supervisorycontroller 42.

Client device 46 may include one or more human-machine interfaces orclient interfaces (e.g., graphical user interfaces, reportinginterfaces, text-based computer interfaces, client-facing web services,web servers that provide pages to web clients, etc.) for controlling,viewing, or otherwise interacting with HVAC system 20, its subsystems,and/or devices. Client device 46 can be a computer workstation, a clientterminal, a remote or local interface, or any other type of userinterface device. Client device 46 can be a stationary terminal or amobile device. For example, client device 46 can be a desktop computer,a computer server with a user interface, a laptop computer, a tablet, asmartphone, a PDA, or any other type of mobile or non-mobile device.

Extremum-Seeking Control Systems With Periodic Dither Signals

Referring now to FIG. 3, a block diagram of an extremum-seeking control(ESC) system 300 with a periodic dither signal is shown, according tosome embodiments. ESC system 300 is shown to include an extremum-seekingcontroller 302 and a plant 304. A plant in control theory is thecombination of a process and one or more mechanically-controlledoutputs. For example, plant 304 can be an air handling unit configuredto control temperature within a building space via one or moremechanically-controlled actuators and/or dampers. In variousembodiments, plant 304 can include a chiller operation process, a damperadjustment process, a mechanical cooling process, a ventilation process,a refrigeration process, or any other process in which an input variableto plant 304 (i.e., manipulated variable u) is adjusted to affect anoutput from plant 304 (i.e., performance variable y).

Extremum-seeking controller 302 uses extremum-seeking control logic tomodulate the manipulated variable u. For example, controller 302 may usea periodic (e.g., sinusoidal) perturbation signal or dither signal toperturb the value of manipulated variable u in order to extract aperformance gradient p. The manipulated variable u can be perturbed byadding periodic oscillations to a DC value of the performance variableu, which may be determined by a feedback control loop. The performancegradient p represents the gradient or slope of the performance variabley with respect to the manipulated variable u. Controller 302 usesextremum-seeking control logic to determine a value for the manipulatedvariable u that drives the performance gradient p to zero.

Controller 302 may determine the DC value of manipulated variable ubased on a measurement or other indication of the performance variable yreceived as feedback from plant 304 via input interface 310.Measurements from plant 304 can include, but are not limited to,information received from sensors about the state of plant 304 orcontrol signals sent to other devices in the system. In someembodiments, the performance variable y is a measured or observedposition of one of valves 92-94. In other embodiments, the performancevariable y is a measured or calculated amount of power consumption, afan speed, a damper position, a temperature, or any other variable thatcan be measured or calculated by plant 304. Performance variable y canbe the variable that extremum-seeking controller 302 seeks to optimizevia an extremum-seeking control technique. Performance variable y can beoutput by plant 304 or observed at plant 304 (e.g., via a sensor) andprovided to extremum-seeking controller at input interface 310.

Input interface 310 provides the performance variable y to performancegradient probe 312 to detect the performance gradient 314. Performancegradient 314 may indicate a slope of the function y=ƒ (u), where yrepresents the performance variable received from plant 304 and urepresents the manipulated variable provided to plant 304. Whenperformance gradient 314 is zero, the performance variable y has anextremum value (e.g., a maximum or minimum). Therefore, extremum-seekingcontroller 302 can optimize the value of the performance variable y bydriving performance gradient 314 to zero.

Manipulated variable updater 316 produces an updated manipulatedvariable u based upon performance gradient 314. In some embodiments,manipulated variable updater 316 includes an integrator to driveperformance gradient 314 to zero. Manipulated variable updater 316 thenprovides an updated manipulated variable u to plant 304 via outputinterface 318. In some embodiments, manipulated variable u is providedto one of dampers 60-64 (FIG. 2) or an actuator affecting dampers 60-64as a control signal via output interface 318. Plant 304 can usemanipulated variable u as a setpoint to adjust the position of dampers60-64 and thereby control the relative proportions of outdoor air 80 andrecirculation air 83 provided to a temperature-controlled space.

Referring now to FIG. 4, a block diagram of another ESC system 400 witha periodic dither signal is shown, according to some embodiments. ESCsystem 400 is shown to include a plant 404 and an extremum-seekingcontroller 402. Controller 402 uses an extremum-seeking control strategyto optimize a performance variable y received as an output from plant404. Optimizing performance variable y can include minimizing y,maximizing y, controlling y to achieve a setpoint, or otherwiseregulating the value of performance variable y.

Plant 404 can be the same as plant 304 or similar to plant 304, asdescribed with reference to FIG. 3. For example, plant 404 can be acombination of a process and one or more mechanically-controlledoutputs. In some embodiments, plant 404 is an air handling unitconfigured to control temperature within a building space via one ormore mechanically-controlled actuators and/or dampers. In otherembodiments, plant 404 can include a chiller operation process, a damperadjustment process, a mechanical cooling process, a ventilation process,or any other process that generates an output based on one or morecontrol inputs.

Plant 404 can be represented mathematically as a combination of inputdynamics 422, a performance map 424, output dynamics 426, anddisturbances d. In some embodiments, input dynamics 422 are lineartime-invariant (LTI) input dynamics and output dynamics 426 are LTIoutput dynamics. Performance map 424 can be a static nonlinearperformance map. Disturbances d can include process noise, measurementnoise, or a combination of both. Although the components of plant 404are shown in FIG. 4, it should be noted that the actual mathematicalmodel for plant 404 does not need to be known in order to apply ESC.

Plant 404 receives a control input u (e.g., a control signal, amanipulated variable, etc.) from extremum-seeking controller 402 viaoutput interface 430. Input dynamics 422 may use the control input u togenerate a function signal x based on the control input (e.g., x=ƒ(u)).Function signal x may be passed to performance map 424 which generatesan output signal z as a function of the function signal (i.e., z=ƒ(x)).The output signal z may be passed through output dynamics 426 to producesignal z′, which is modified by disturbances d to produce performancevariable y (e.g., y=z′+d). Performance variable y is provided as anoutput from plant 404 and received at extremum-seeking controller 402.Extremum-seeking controller 402 may seek to find values for x and/or uthat optimize the output z of performance map 424 and/or the performancevariable y.

Still referring to FIG. 4, extremum-seeking controller 402 is shownreceiving performance variable y via input interface 432 and providingperformance variable y to a control loop 405 within controller 402.Control loop 405 is shown to include a high-pass filter 406, ademodulation element 408, a low-pass filter 410, an integrator feedbackcontroller 412, and a dither signal element 414. Control loop 405 may beconfigured to extract a performance gradient p from performance variabley using a dither-demodulation technique. Integrator feedback controller412 analyzes the performance gradient p and adjusts the DC value of theplant input (i.e., the variable w) to drive performance gradient p tozero.

The first step of the dither-demodulation technique is performed bydither signal generator 416 and dither signal element 414. Dither signalgenerator 416 generates a periodic dither signal v, which is typically asinusoidal signal. Dither signal element 414 receives the dither signalv from dither signal generator 416 and the DC value of the plant input wfrom controller 412. Dither signal element 414 combines dither signal vwith the DC value of the plant input w to generate the perturbed controlinput u provided to plant 404 (e.g., u=w+v). The perturbed control inputu is provided to plant 404 and used by plant 404 to generate performancevariable y as previously described.

The second step of the dither-demodulation technique is performed byhigh-pass filter 406, demodulation element 408, and low-pass filter 410.High-pass filter 406 filters the performance variable y and provides thefiltered output to demodulation element 408. Demodulation element 408demodulates the output of high-pass filter 406 by multiplying thefiltered output by the dither signal v with a phase shift 418 applied.The DC value of this multiplication is proportional to the performancegradient p of performance variable y with respect to the control inputu. The output of demodulation element 408 is provided to low-pass filter410, which extracts the performance gradient p (i.e., the DC value ofthe demodulated output). The estimate of the performance gradient p isthen provided to integrator feedback controller 412, which drives theperformance gradient estimate p to zero by adjusting the DC value w ofthe plant input u.

Still referring to FIG. 4, extremum-seeking controller 402 is shown toinclude an amplifier 420. It may be desirable to amplify the dithersignal v such that the amplitude of the dither signal v is large enoughfor the effects of dither signal v to be evident in the plant output y.The large amplitude of dither signal v can result in large variations inthe control input u, even when the DC value w of the control input uremains constant. Graphs illustrating a control input u and aperformance variable y with periodic oscillations caused by a periodicdither signal v are shown in FIGS. 6A-6B (described in greater detailbelow). Due to the periodic nature of the dither signal v, the largevariations in the plant input u (i.e., the oscillations caused by thedither signal v) are often noticeable to plant operators.

Additionally, it may be desirable to carefully select the frequency ofthe dither signal v to ensure that the ESC strategy is effective. Forexample, it may be desirable to select a dither signal frequency ω_(v)based on the natural frequency ω_(n) of plant 304 to enhance the effectof the dither signal v on the performance variable y. It can bedifficult and challenging to properly select the dither frequency ω_(v)without knowledge of the dynamics of plant 404. For these reasons, theuse of a periodic dither signal v is one of the drawbacks of traditionalESC.

In ESC system 400, the output of high-pass filter 406 can be representedas the difference between the value of the performance variable y andthe expected value of the performance variable y, as shown in thefollowing equation:

Output of High-Pass Filter: y−E[y]

where the variable E[y] is the expected value of the performancevariable y. The result of the cross-correlation performed bydemodulation element 408 (i.e., the output of demodulation element 408)can be represented as the product of the high-pass filter output and thephase-shifted dither signal, as shown in the following equation:

Result of Cross-Correlation: (y−E[y])(v−E[v])

where the variable E [v] is the expected value of the dither signal v.The output of low-pass filter 410 can be represented as the covarianceof the dither signal v and the performance variable y, as shown in thefollowing equation:

Output of Low-Pass Filter: E[(y−E[y])(v−E[u])]≡Cov(v,y)

where the variable E[u] is the expected value of the control input u.

The preceding equations show that ESC system 400 generates an estimatefor the covariance Cov(v,y) between the dither signal v and the plantoutput (i.e., the performance variable y). The covariance Cov(v,y) canbe used in ESC system 400 as a proxy for the performance gradient p. Forexample, the covariance Cov(v,y) can be calculated by high-pass filter406, demodulation element 408, and low-pass filter 410 and provided as afeedback input to integrator feedback controller 412. Integratorfeedback controller 412 can adjust the DC value w of the plant input uin order to minimize the covariance Cov(v,y) as part of the feedbackcontrol loop.

Extremum-Seeking Control System with Stochastic Excitation Signal

Referring now to FIG. 5, a block diagram of an ESC system 500 with astochastic excitation signal is shown, according to some embodiments.ESC system 500 is shown to include a plant 504 and an extremum-seekingcontroller 502. Controller 502 is shown receiving a performance variabley as feedback from plant 504 via input interface 526 and providing acontrol input u to plant 504 via output interface 524. Controller 502may operate in a manner similar to controllers 302 and 402, as describedwith reference to FIGS. 3-4. For example, controller 502 can use anextremum-seeking control (ESC) strategy to optimize the performancevariable y received as an output from plant 504. However, rather thanperturbing the control input u with a periodic dither signal, controller502 may perturb the control input u with a stochastic excitation signalq. Controller 502 can adjust the control input u to drive the gradientof performance variable y to zero. In this way, controller 502identifies values for control input u that achieve an optimal value(e.g., a maximum or a minimum) for performance variable y.

In some embodiments, the ESC logic implemented by controller 502generates values for control input u based on a received control signal(e.g., a setpoint, an operating mode signal, etc.). The control signalmay be received from a user control (e.g., a thermostat, a local userinterface, etc.), client devices 536 (e.g., computer terminals, mobileuser devices, cellular phones, laptops, tablets, desktop computers,etc.), a supervisory controller 532, or any other external system ordevice. In various embodiments, controller 502 can communicate withexternal systems and devices directly (e.g., using NFC, Bluetooth, WiFidirect, cables, etc.) or via a communications network 534 (e.g., aBACnet network, a LonWorks network, a LAN, a WAN, the Internet, acellular network, etc.) using wired or wireless electronic datacommunications

Plant 504 can be similar to plant 404, as described with reference toFIG. 4. For example, plant 504 can be a combination of a process and oneor more mechanically-controlled outputs. In some embodiments, plant 504is an air handling unit configured to control temperature within abuilding space via one or more mechanically-controlled actuators and/ordampers. In other embodiments, plant 404 can include a chiller operationprocess, a damper adjustment process, a mechanical cooling process, aventilation process, or any other process that generates an output basedon one or more control inputs.

Plant 504 can be represented mathematically as a static nonlinearity inseries with a dynamic component. For example, plant 504 is shown toinclude a static nonlinear function block 516 in series with a constantgain block 518 and a transfer function block 520. Although thecomponents of plant 504 are shown in FIG. 5, it should be noted that theactual mathematical model for plant 504 does not need to be known inorder to apply ESC. Plant 504 receives a control input u (e.g., acontrol signal, a manipulated variable, etc.) from extremum-seekingcontroller 502 via output interface 524. Nonlinear function block 516can use the control input u to generate a function signal x based on thecontrol input (e.g., x=ƒ(u)). Function signal x can be passed toconstant gain block 518, which multiplies the function signal x by theconstant gain K to generate the output signal z (i.e., z=Kx). The outputsignal z can be passed through transfer function block 520 to producesignal z′, which is modified by disturbances d to produce performancevariable y (e.g., y=z′+d). Disturbances d can include process noise,measurement noise, or a combination of both. Performance variable y isprovided as an output from plant 504 and received at extremum-seekingcontroller 502.

Still referring to FIG. 5, controller 502 is shown to include acommunications interface 530, an input interface 526, and an outputinterface 524. Interfaces 530 and 524-526 can include any number ofjacks, wire terminals, wire ports, wireless antennas, or othercommunications interfaces for communicating information and/or controlsignals. Interfaces 530 and 524-526 can be the same type of devices ordifferent types of devices. For example, input interface 526 can beconfigured to receive an analog feedback signal (e.g., an outputvariable, a measured signal, a sensor output, a controlled variable)from plant 504, whereas communications interface 530 can be configuredto receive a digital setpoint signal from supervisory controller 532 vianetwork 534. Output interface 524 can be a digital output (e.g., anoptical digital interface) configured to provide a digital controlsignal (e.g., a manipulated variable, a control input) to plant 504. Inother embodiments, output interface 524 is configured to provide ananalog output signal.

In some embodiments interfaces 530 and 524-526 can be joined as one ortwo interfaces rather than three separate interfaces. For example,communications interface 530 and input interface 526 can be combined asone Ethernet interface configured to receive network communications fromsupervisory controller 532. In some embodiments, supervisory controller532 provides both a setpoint and feedback via an Ethernet network (e.g.,network 534). In such an embodiment, output interface 524 may bespecialized for a controlled component of plant 504. In otherembodiments, output interface 524 can be another standardizedcommunications interface for communicating data or control signals.Interfaces 530 and 524-526 can include communications electronics (e.g.,receivers, transmitters, transceivers, modulators, demodulators,filters, communications processors, communication logic modules,buffers, decoders, encoders, encryptors, amplifiers, etc.) configured toprovide or facilitate the communication of the signals described herein.

Still referring to FIG. 5, controller 502 is shown to include aprocessing circuit 538 having a processor 540 and memory 542. Processor540 can be a general purpose or specific purpose processor, anapplication specific integrated circuit (ASIC), one or more fieldprogrammable gate arrays (FPGAs), a group of processing components, orother suitable processing components. Processor 540 is configured toexecute computer code or instructions stored in memory 542 or receivedfrom other computer readable media (e.g., CDROM, network storage, aremote server, etc.).

Memory 542 can include one or more devices (e.g., memory units, memorydevices, storage devices, etc.) for storing data and/or computer codefor completing and/or facilitating the various processes described inthe present disclosure. Memory 542 can include random access memory(RAM), read-only memory (ROM), hard drive storage, temporary storage,non-volatile memory, flash memory, optical memory, or any other suitablememory for storing software objects and/or computer instructions. Memory542 can include database components, object code components, scriptcomponents, or any other type of information structure for supportingthe various activities and information structures described in thepresent disclosure. Memory 542 can be communicably connected toprocessor 540 via processing circuit 538 and can include computer codefor executing (e.g., by processor 540) one or more processes describedherein.

Still referring to FIG. 5, extremum-seeking controller 502 is shownreceiving performance variable y via input interface 526 and providingperformance variable y to a control loop 505 within controller 502.Control loop 505 is shown to include a recursive gradient estimator 506,a feedback controller 508, and an excitation signal element 510. Controlloop 505 may be configured to determine the gradient

$\frac{dy}{du}$

of the performance variable y with respect to the control input u and toadjust the DC value of the control input u (i.e., the variable w) todrive the gradient

$\frac{dy}{du}$

to zero.

Recursive Gradient Estimation

Recursive gradient estimator 506 can be configured to estimate thegradient

$\frac{dy}{du}$

of the performance variable y with respect to the control input u. Thegradient

$\frac{dy}{du}$

may be similar to the performance gradient p determined in ESC system400. However, the fundamental difference between ESC system 500 and ESCsystem 400 is the way that the gradient

$\frac{dy}{du}$

is obtained. In ESC system 400, the performance gradient p is obtainedvia the dither-demodulation technique described with reference to FIG.4, which is analogous to covariance estimation. Conversely, the gradient

$\frac{dy}{du}$

in ESC system 500 is obtained by performing a recursive regressiontechnique to estimate the slope of the performance variable y withrespect to the control input u. The recursive estimation technique maybe performed by recursive gradient estimator 506.

Recursive gradient estimator 506 can use any of a variety of recursiveestimation techniques to estimate the gradient

$\frac{dy}{du}.$

For example, recursive gradient estimator 506 can use a recursiveleast-squares (RLS) estimation technique to generate an estimate of thegradient

$\frac{dy}{du}.$

In some embodiments, recursive gradient estimator 506 uses exponentialforgetting as part of the RLS estimation technique. Exponentialforgetting reduces the required amount of data storage relative to batchprocessing. Exponential forgetting also allows the RLS estimationtechnique to remain more sensitive to recent data and thus moreresponsive to a shifting optimal point. An example a RLS estimationtechnique which can be performed recursive gradient estimator 506 isdescribed in detail below.

Recursive gradient estimator 506 is shown receiving the performancevariable y from plant 504 and the control input u from excitation signalelement 510. In some embodiments, recursive gradient estimator 506receives multiple samples or measurements of the performance variable yand the control input u over a period of time. Recursive gradientestimator 506 can use a sample of the control input u at time k toconstruct an input vector x_(k) as shown in the following equation:

$x_{k} = \begin{bmatrix}1 \\u_{k\;}\end{bmatrix}$

where u_(k) is the value of the control input u at time k. Similarly,recursive gradient estimator 506 can construct a parameter vector{circumflex over (θ)}_(k) as shown in the following equation:

${\hat{\theta}}_{k} = \begin{bmatrix}{\hat{\theta}}_{1} \\{\hat{\theta}}_{2}\end{bmatrix}$

where the parameter {circumflex over (θ)}₂ is the estimate of thegradient

$\frac{dy}{du}$

at time k.

Recursive gradient estimator 506 can estimate the performance variableŷ_(k) at time k using the following linear model:

ŷ _(k) =x _(k) ^(T){circumflex over (θ)}_(k-1)

The prediction error of this model is the difference between the actualvalue of the performance variable y_(k) at time k and the estimatedvalue of the performance variable ŷ_(k) at time k as shown in thefollowing equation:

e _(k) =y _(k) −ŷ _(k) =y _(k) −x _(k) ^(T){circumflex over (θ)}_(k-1)

Recursive gradient estimator 506 can use the estimation error e_(k) inthe RLS technique to determine the parameter values {circumflex over(θ)}_(k). Any of a variety of RLS techniques can be used in variousimplementations. An example of a RLS technique which can be performed byrecursive gradient estimator 506 is as follows:

g _(k) =P _(k-1) x _(k)(λ+x _(k) ^(T) P _(k-1) x _(k))⁻¹

P _(k)=λ⁻¹ P _(k-1) −g _(j) x _(k) ^(T)λ⁻¹ P _(k-1)

{circumflex over (θ)}_(k)={circumflex over (θ)}_(k-1) +e _(k) g _(k)

where g_(k) is a gain vector, P_(k) is a covariance matrix, and A is aforgetting factor (λ<1). In some embodiments, the forgetting factor λ isdefined as follows:

$\lambda = e^{- \frac{\Delta \; t}{\tau}}$

where Δt is the sampling period and τ is the forgetting time constant.

Recursive gradient estimator 506 can use the equation for g_(k) tocalculate the gain vector g_(k) at time k based on a previous value ofthe covariance matrix P_(k-1) at time k−1, the value of the input vectorx_(k) ^(T) at time k, and the forgetting factor. Recursive gradientestimator 506 can use the equation for P_(k) to calculate the covariancematrix P_(k) at time k based on the forgetting factor λ, the value ofthe gain vector g_(k) at time k, and the value of the input vector x_(k)^(T) at time k. Recursive gradient estimator 506 can use the equationfor {circumflex over (θ)}_(k) to calculate the parameter vector{circumflex over (θ)}_(k) at time k based on the error e_(k) at time kand the gain vector g_(k) at time k. Once the parameter vector{circumflex over (θ)}_(k) is calculated, recursive gradient estimator506 can determine the value of the gradient

$\frac{dy}{du}$

by extracting the value of the {circumflex over (θ)}₂ parameter from{circumflex over (θ)}_(k), as shown in the following equations:

${{\hat{\theta}}_{k} = \begin{bmatrix}{\hat{\theta}}_{1} \\{\hat{\theta}}_{2}\end{bmatrix}},{\frac{dy}{du} = {\hat{\theta}}_{2}}$

In various embodiments, recursive gradient estimator 506 can use any ofa variety of other recursive estimation techniques to estimate

$\frac{dy}{du}.$

For example, recursive gradient estimator 506 can use a Kalman filter, anormalized gradient technique, an unnormalized gradient adaptiontechnique, a recursive Bayesian estimation technique, or any of avariety of linear or nonlinear filters to estimate

$\frac{dy}{du}.$

In other embodiments, recursive gradient estimator 506 can use a batchestimation technique rather than a recursive estimation technique. Assuch, gradient estimator 506 can be a batch gradient estimator ratherthan a recursive gradient estimator. In a batch estimation technique,gradient estimator 506 can use a batch of previous values for thecontrol input u and the performance variable y (e.g., a vector or set ofprevious or historical values) as inputs to a batch regressionalgorithm. Suitable regression algorithms may include, for example,ordinary least squares regression, polynomial regression, partial leastsquares regression, ridge regression, principal component regression, orany of a variety of linear or nonlinear regression techniques.

In some embodiments, it is desirable for recursive gradient estimator506 to use a recursive estimation technique rather than a batchestimation technique due to several advantages provided by the recursiveestimation technique. For example, the recursive estimation techniquedescribed above (i.e., RLS with exponential forgetting) has been shownto greatly improve the performance of the gradient estimation techniquerelative to batch least-squares. In addition to requiring less datastorage than batch processing, the RLS estimation technique withexponential forgetting can remain more sensitive to recent data and thusmore responsive to a shifting optimal point.

In some embodiments, recursive gradient estimator 506 estimates thegradient

$\frac{dy}{du}$

using the covariance between the control input u and the performancevariable y. For example, the estimate of the slope {circumflex over (β)}in a least-squares approach can be defined as:

$\hat{\beta} = \frac{{Cov}\left( {u,y} \right)}{{Var}(u)}$

where Cov(u,y) is the covariance between the control input u and theperformance variable y, and Var(u) is the variance of the control inputu. Recursive gradient estimator 506 can calculate the estimated slope{circumflex over (β)} using the previous equation and use the estimatedslope {circumflex over (β)} as a proxy for the gradient

$\frac{dy}{du}.$

Notably, the estimated slope {circumflex over (β)} is a function of onlythe control input u and the performance variable y. This is differentfrom the covariance derivation technique described with reference toFIG. 4 in which the estimated performance gradient p was a function ofthe covariance between the dither signal v and the performance variabley. By replacing the dither signal v with the control input u, controller502 can generate an estimate for the slope {circumflex over (β)} withoutany knowledge of the dither signal v (shown in FIG. 4) or the excitationsignal q (shown in FIG. 5).

In some embodiments, recursive gradient estimator 506 uses ahigher-order model (e.g., a quadratic model, a cubic model, etc.) ratherthan a linear model to estimate the performance variable ŷ_(k). Forexample, recursive gradient estimator 506 can estimate the performancevariable ŷ_(k) at time k using the following quadratic model:

ŷ _(k)={circumflex over (θ)}₁+{circumflex over (θ)}₂ u _(k)+{circumflexover (θ)}₃ u _(k) ²+∈_(k)

which can be written in the form ŷ_(k)=x_(k) ^(T){circumflex over(θ)}_(k-1) by updating the input vector x_(k) and the parameter vector{circumflex over (θ)}_(k) as follows:

$x_{k} = \begin{bmatrix}1 \\u_{k} \\u_{k}^{2}\end{bmatrix}$ ${\hat{\theta}}_{k} = \begin{bmatrix}{\hat{\theta}}_{1} \\{\hat{\theta}}_{2} \\{\hat{\theta}}_{3}\end{bmatrix}$

Recursive gradient estimator 506 can use the quadratic model to fit aquadratic curve (rather than a straight line) to the data points definedby combinations of the control input u and the performance variable y atvarious times k. The quadratic model provides second-order informationnot provided by the linear model and can be used to improve theconvergence of feedback controller 508. For example, with a linearmodel, recursive gradient estimator 506 can calculate the gradient

$\frac{dy}{du}$

at a particular location along the curve (i.e., for a particular valueof the control input u) and can provide the gradient

$\frac{dy}{du}$

as a feedback signal. For embodiments that use a linear model toestimate ŷ_(k), the gradient

$\frac{dy}{du}$

(i.e., the derivative of the linear model with respect to u) is a scalarvalue. When controller 508 receives a scalar value for the gradient

$\frac{dy}{du}$

as a feedback signal, controller 508 can incrementally adjust the valueof the control input u in a direction that drives the gradient

$\frac{dy}{du}$

toward zero until the optimal value of the control input u is reached(i.e., the value of the control input u that results in the gradient

$\left. {\frac{dy}{du} = 0} \right).$

With a quadratic model, recursive gradient estimator 506 can providefeedback controller 508 with a function for the gradient

$\frac{dy}{du}$

rather than a simple scalar value. For embodiments that use a quadraticmodel to estimate ŷ_(k), the gradient

$\frac{dy}{du}$

(i.e., the derivative of the quadratic model with respect to u) is alinear function of the control input u

$\left( {{e.g.},{\frac{dy}{du} = {{2{\hat{\theta}}_{3}u_{k}} + {\hat{\theta}}_{2}}}} \right).$

When controller 508 receives a linear function for the gradient

$\frac{dy}{du}$

as a feedback signal, controller 508 can analytically calculate theoptimal value of the control input u that will result in the gradient

$\frac{dy}{du} = 0$$\left( {{e.g.},{u_{k,{opt}} = {- \frac{{\hat{\theta}}_{2}}{2{\hat{\theta}}_{3}}}}} \right).$

Accordingly, controller 508 can adjust the control input u using smartsteps that rapidly approach the optimal value without relying onincremental adjustment and experimentation to determine whether thegradient

$\frac{dy}{du}$

is moving toward zero.

Stochastic Excitation Signal

Still referring to FIG. 5, extremum-seeking controller 502 is shown toinclude a stochastic signal generator 512 and an integrator 514. Inorder to estimate the gradient

$\frac{dy}{du}$

reliably, it may be desirable to provide sufficient variation in thecontrol input u that carries through to the performance variable y.Controller 502 can use stochastic signal generator 512 and integrator514 to generate a persistent excitation signal q. The excitation signalq can be added to the DC value w of the control input u at excitationsignal element 510 to form the control input u (e.g., u=w+q).

Stochastic signal generator 512 can be configured to generate astochastic signal. In various embodiments, the stochastic signal can bea random signal (e.g., a random walk signal, a white noise signal,etc.), a non-periodic signal, an unpredictable signal, a disturbancesignal, or any other type of non-deterministic or non-repeating signal.In some embodiments, the stochastic signal has a non-zero mean. Thestochastic signal can be integrated by integrator 514 to generate theexcitation signal q.

Excitation signal q can provide variation in the control input usufficient for the gradient estimation technique performed by recursivegradient estimator 506. In some instances, the addition of excitationsignal q causes the control input u to drift away from its optimumvalue. However, feedback controller 508 can compensate for such drift byadjusting the DC value w such that the control input u is continuouslypulled back toward its optimum value. As with traditional ESC, themagnitude of the excitation signal q can be selected (e.g., manually bya user or automatically by controller 502) to overcome any additivenoise found in the performance variable y (e.g., process noise,measurement noise, etc.).

The stochastic excitation signal q generated by extremum-seekingcontroller 502 has several advantages over the periodic dither signal vgenerated by controller 402. For example, the stochastic excitationsignal q is less perceptible than a traditional periodic dither signalv. As such, the effects of the stochastic excitation signal q on thecontrol input u are less noticeable than the periodic oscillationscaused by the traditional periodic dither signal v. Graphs illustratinga control input u excited by the stochastic excitation signal q and theresulting performance variable y are shown in FIGS. 7A-7B (described ingreater detail below).

Another advantage of the stochastic excitation signal q is that tuningcontroller 502 is simpler because the dither frequency ω_(v) is nolonger a required parameter. Accordingly, controller 502 does not needto know or estimate the natural frequency of plant 504 when generatingthe stochastic excitation signal q. In some embodiments,extremum-seeking controller 502 provides multiple control inputs u toplant 504. Each of the control inputs can be excited by a separatestochastic excitation signal q. Since each of the stochastic excitationsignals q is random, there is no need to ensure that the stochasticexcitation signals q are not correlated with each other. Controller 502can calculate the gradient

$\frac{dy}{du}$

of the performance variable y with respect to each of the control inputsu without performing a frequency-specific dither-demodulation technique.

Correlation Coefficient

One of the problems with traditional ESC is that the performancegradient

$\frac{dy}{du}$

is a function of the range or scale of the performance variable y. Therange or scale of the performance variable y can depend on the staticand dynamic components of plant 504. For example, plant 504 is shown toinclude a nonlinear function ƒ(u) (i.e., function block 516) in serieswith a constant gain K (i.e., constant gain block 518). It is apparentfrom this representation that the range or scale of the performancevariable y is a function of the constant gain K.

The value of the performance gradient

$\frac{dy}{du}$

may vary based on the value of the control input u due to thenonlinearity provided by the nonlinear function ƒ(u). However, the scaleof the performance gradient

$\frac{dy}{du}$

is also dependent upon the value of the constant gain K. For example,the performance gradient

$\frac{dy}{du}$

can be determined using the following equation:

$\frac{dy}{du} = {{Kf}^{\prime}(u)}$

where K is the constant gain and ƒ′(u) is the derivative of the functionƒ(u). It can be desirable to scale or normalize the performance gradient

$\frac{dy}{du}$

(e.g., by multiplying by a scaling parameter κ) in order to provideconsistent feedback control loop performance. However, without knowledgeof the scale of the performance variable y (e.g., without knowing theconstant gain K applied by plant 504), it can be challenging todetermine an appropriate value for the scaling parameter κ.

Still referring to FIG. 5, extremum-seeking controller 502 is shown toinclude a correlation coefficient estimator 528. Correlation coefficientestimator 528 can be configured to generate a correlation coefficient ρand provide the correlation coefficient ρ to feedback controller 508.The correlation coefficient ρ can be related to the performance gradient

$\frac{dy}{du}\left( {{e.g.},{{proportional}\mspace{14mu} {to}\mspace{14mu} \frac{dy}{du}}} \right)$

but scaled based on the range of the performance variable y. Forexample, the correlation coefficient ρ can be a normalized measure ofthe performance gradient

$\frac{dy}{du}$

(e.g., scaled to the range 0≤ρ≤1).

Correlation coefficient estimator 528 is shown receiving the controlinput u and the performance variable y as inputs. Correlationcoefficient estimator 528 can generate the correlation coefficient ρbased on the variance and covariance of the control input u and theperformance variable y, as shown in the following equation:

$\rho = \frac{{Cov}\left( {u,y} \right)}{\sqrt{{{Var}(u)}{{Var}(y)}}}$

where Cov(u,y) is the covariance between the control input u and theperformance variable y, Var(u) is the variance of the control input u,and Var(y) is the variance of the performance variable y. The previousequation can be rewritten in terms of the standard deviation σ_(u) ofthe control input u and the standard deviation σ_(y) of the performancevariable y as follows:

$\rho = \frac{{Cov}\left( {u,y} \right)}{\sigma_{u}\sigma_{y}}$

where Var(u)=σ_(u) ² and Var(y)=σ_(y) ²

In some embodiments, correlation coefficient estimator 528 estimates thecorrelation coefficient ρ using a recursive estimation technique. Forexample, correlation coefficient estimator 528 can calculateexponentially-weighted moving averages (EWMAs) of the control input uand the performance variable y using the following equations:

${\overset{\_}{u}}_{k} = {{\overset{\_}{u}}_{k - 1} + \frac{u_{k} - {\overset{\_}{u}}_{k - 1}}{\min \left( {k,W} \right)}}$${\overset{\_}{y}}_{k} = {{\overset{\_}{y}}_{k - 1} + \frac{y_{k} - {\overset{\_}{y}}_{k - 1}}{\min \left( {k,W} \right)}}$

where û_(k) and ŷ_(k) are the EWMAs of the control input u and theperformance variable y at time k, ū_(k-1) and y _(k-1) are the previousEWMAs of the control input u and the performance variable y at time k−1,u_(k) and y_(k) are the current values of the control input u and theperformance variable y at time k, k is the total number of samples thathave been collected of each variable, and W is the duration of theforgetting window.

Similarly, correlation coefficient estimator 528 can calculate EWMAs ofthe control input variance Var(u), the performance variable varianceVar(y), and the covariance Cov(u,y) using the following equations:

$V_{u,k} = {V_{u,{k - 1}} + \frac{\left( {u_{k} - {\overset{\_}{u}}_{k}} \right)^{2} - V_{u,{k - 1}}}{\min \left( {k,W} \right)}}$$V_{y,k} = {V_{y,{k - 1}} + \frac{\left( {y_{k} - {\overset{\_}{y}}_{k}} \right)^{2} - V_{y,{k - 1}}}{\min \left( {k,W} \right)}}$

where V_(u,k), V_(y,k), and c_(k) are the EWMAs of the control inputvariance Var(u), the performance variable variance Var(y), and thecovariance Cov(u,y), respectively, at time k. V_(u,k-1), V_(y,k-1), andc_(k-1) are the EWMAs of the control input variance Var(u), theperformance variable variance Var(y), and the covariance Cov(u,y),respectively, at time k−1. Correlation coefficient estimator 528 cangenerate an estimate of the correlation coefficient ρ based on theserecursive estimates using the following equation:

${\hat{\rho}}_{k} = \frac{c_{k}}{\sqrt{V_{u,k}V_{y,k}}}$

In some embodiments, correlation coefficient estimator 528 generates thecorrelation coefficient ρ based on the estimated slope {circumflex over(β)}. As previously described, the estimated slope {circumflex over (β)}can be calculated using the following equation:

$\hat{\beta} = {\frac{{Cov}\left( {u,y} \right)}{{Var}(u)} = \frac{{Cov}\left( {u,y} \right)}{\sigma_{u}^{2}}}$

where Cov(u,y) is the covariance between the control input u and theperformance variable y, and Var(u) is the variance of the control inputu (i.e., σ_(u) ²). Correlation coefficient estimator 528 can calculatethe correlation coefficient ρ from the slope β using the followingequation:

$\rho = {\hat{\beta}\frac{\sigma_{u}}{\sigma_{y}}}$

From the previous equation, it can be seen that the correlationcoefficient ρ and the estimated slope {circumflex over (β)} are equalwhen the standard deviations σ_(u) and σ_(y) are equal (i.e., whenσ_(u)=σ_(y)).

Correlation coefficient estimator 528 can receive the estimated slope{circumflex over (β)} from recursive gradient estimator 506 or calculatethe estimated slope {circumflex over (β)} using a set of values for thecontrol input u and the performance variable y. For example, with theassumption of finite variance in u and y, correlation coefficientestimator 528 can estimate the slope {circumflex over (β)} using thefollowing least squares estimation:

$\hat{\beta} = {\left( {\sum\limits_{i = {t - N}}^{t}{u_{i}u_{i}^{T}}} \right)^{- 1}\left( {\sum\limits_{i = {t - N}}^{t}{u_{i}y_{i}}} \right)}$

For a small range of the control input u, the estimated slope{circumflex over (β)} can be used as a proxy for the performancegradient, as shown in the following equation:

$\hat{\beta} = {\frac{dy}{du}{{Kf}^{\; \prime}(u)}}$

As shown in the previous equation, the estimated slope {circumflex over(β)} contains the constant gain K, which may be unknown. However,normalization provided by the standard deviations σ_(u) and σ_(y)cancels the effect of the constant gain K. For example, the standarddeviation σ_(y) of the performance variable y is related to the standarddeviation σ_(u) of the control input u as shown in the followingequations:

σ_(y) = K σ_(u) $\frac{\sigma_{u}}{\sigma_{y}} = \frac{1}{K}$

Multiplying the estimated slope {circumflex over (β)} by the ratio

$\frac{\sigma_{u}}{\sigma_{y}}$

to calculate the correlation coefficient ρ is equivalent to dividing bythe constant gain K. Both the correlation coefficient ρ and theestimated slope {circumflex over (β)} indicate the strength of therelationship between the control input u and the performance variable y.However, the correlation coefficient ρ has the advantage of beingnormalized which makes tuning the feedback control loop much simpler.

In some embodiments, the correlation coefficient ρ is used by feedbackcontroller 508 instead of the performance gradient

$\frac{dy}{du}.$

For example, feedback controller 508 can adjust the DC value w of thecontrol input u to drive the correlation coefficient ρ to zero. Oneadvantage of using the correlation coefficient ρ in place of theperformance gradient

$\frac{dy}{du}$

is that the tuning parameters used by feedback controller 508 can be ageneral set of tuning parameters which do not need to be customized oradjusted based on the scale of the performance variable y. Thisadvantage eliminates the need to perform control-loop-specific tuningfor feedback controller 508 and allows feedback controller 508 to use ageneral set of tuning parameters that are applicable across manydifferent control loops and/or plants.

Example Graphs

Referring now to FIGS. 6A-7B, several graphs 600-750 comparing theperformance of extremum-seeking controller 402 and extremum-seekingcontroller 502 are shown, according to some embodiments. Controllers 402and 502 were used to control a dynamic system that has an optimalcontrol input value of u=2 and an optimal performance variable of y=−10.Both controllers 402 and 502 were started at a value of u=4 and allowedto adjust the value of the control input u using the extremum-seekingcontrol techniques described with reference to FIGS. 4-5. Controller 402uses a periodic dither signal v, whereas controller 502 uses astochastic excitation signal q.

Referring particularly to FIGS. 6A-6B, graphs 600 and 650 illustrate theperformance of extremum-seeking controller 402, as described withreference to FIG. 4. Controller 402 uses a periodic dither signal v toperturb the control input u. Graph 600 shows the value of the controlinput u at various sample times, whereas graph 650 shows correspondingvalue of the performance variable y. The control input u starts at avalue of u=4 and is perturbed using a periodic (i.e., sinusoidal) dithersignal v. The oscillatory perturbation caused by the periodic dithersignal v is visible in both the control input u and the performancevariable y.

Referring particularly to FIGS. 7A-7B, graphs 700 and 750 illustrate theperformance of extremum-seeking controller 502, as described withreference to FIG. 5. Controller 502 uses a stochastic excitation signalq to perturb the control input u. Graph 700 shows the value of thecontrol input u at various sample times, whereas graph 750 showscorresponding value of the performance variable y. The control input ustarts at a value of u=4 and is perturbed using a stochastic excitationsignal q. The stochastic excitation signal q applies a random walk tothe control input u. However, since the stochastic excitation signal qis non-periodic and effective small amplitudes, the perturbation causedby the stochastic excitation signal q is barely discernable in graphs700 and 750. Additionally, control input u in graph 700 reaches theoptimal value quicker than the control input in graph 600.

Extremum-Seeking Control Techniques

Referring now to FIG. 8, a flow diagram 800 illustrating anextremum-seeking control (ESC) technique is shown, according to someembodiments. The ESC technique shown in flow diagram 800 can beperformed by one or more components of a feedback controller (e.g.,controller 502) to monitor and control a plant (e.g., plant 504). Forexample, controller 502 can use the ESC technique to determine anoptimal value of a control input u provided to plant 504 by perturbingthe control input u with a stochastic excitation signal q.

Flow diagram 800 is shown to include providing a control input u to aplant (block 802) and receiving a performance variable y as a feedbackfrom a plant (block 804). The control input u can be provided by anextremum-seeking controller and/or a feedback controller for the plant.The controller can be any of the controllers previously described (e.g.,controller 302, controller 402, controller 502, etc.) or any other typeof controller that provides a control input u to a plant. In someembodiments, the controller is an extremum-seeking controller configuredto achieve an optimal value for the performance variable y by adjustingthe control input u. The optimal value can be an extremum (e.g., amaximum or a minimum) of the performance variable y.

A plant in control theory is the combination of a process and one ormore mechanically-controlled outputs. The plant can be any of the plantspreviously described (e.g., plant 304, plant 404, plant 504, etc.) orany other controllable system or process. For example, the plant can bean air handling unit configured to control temperature within a buildingspace via one or more mechanically-controlled actuators and/or dampers.In various embodiments, the plant can include a chiller operationprocess, a damper adjustment process, a mechanical cooling process, aventilation process, a refrigeration process, or any other process inwhich a control input u to the plant is adjusted to affect theperformance variable y. The performance variable y can be a measuredvariable observed by one or more sensors of the plant (e.g., a measuredtemperature, a measured power consumption, a measured flow rate, etc.),a calculated variable based on measured or observed values (e.g., acalculated efficiency, a calculated power consumption, a calculatedcost, etc.) or any other type of variable that indicates the performanceof the plant in response to the control input u.

Flow diagram 800 is shown to include estimating a gradient of theperformance variable y with respect to the control input u (block 806).In some embodiments, the gradient is the performance gradient pdescribed with reference to FIG. 4. In other embodiments, the gradientcan be the performance gradient

$\frac{dy}{du}$

or the estimated slope {circumflex over (β)} described with reference toFIG. 5. For example, the gradient can be a slope or derivative of acurve defined by the function y=ƒ(u) at a particular location along thecurve (e.g., at a particular value of u). The gradient can be estimatedusing one or more pairs of values for the control input u and theperformance variable y.

In some embodiments, the gradient is estimated by performing a recursivegradient estimation technique. The recursive gradient estimationtechnique may include obtaining a model for the performance variable yas a function of the control input u. For example, the gradient can beestimated using the following linear model:

ŷ _(k) =x _(k) ^(T){circumflex over (θ)}_(k-1)

where x_(k) is an input vector and {circumflex over (θ)}_(k) is aparameter vector. The input vector x_(k) and the parameter vector{circumflex over (θ)}_(k) can be defined as follows:

$x_{k} = {{\begin{bmatrix}1 \\u_{k}\end{bmatrix}\mspace{20mu} {\hat{\theta}}_{k}} = \begin{bmatrix}{\hat{\theta}}_{1} \\{\hat{\theta}}_{2}\end{bmatrix}}$

where u_(k) is the value of the control input u at time k and theparameter {circumflex over (θ)}₂ is the estimate of the gradient

$\frac{dy}{du}$

at time k.

The prediction error of this model is the difference between the actualvalue of the performance variable y_(k) at time k and the estimatedvalue of the performance variable ŷ_(k) at time k as shown in thefollowing equation:

e _(k) =y _(k) =ŷ _(k) =y _(k) −x _(k) ^(T){circumflex over (θ)}_(k-1)

The estimation error e_(k) can be used in the recursive gradientestimation technique to determine the parameter values {circumflex over(θ)}_(k). Any of a variety of regression techniques can be used toestimate values for the parameter vector {circumflex over (θ)}_(k).

In some embodiments, a higher-order model (e.g., a quadratic model, acubic model, etc.) rather than a linear model can be used to estimatethe gradient. For example, the following quadratic model can be used toestimate the gradient

$\frac{dy}{du}$

at a particular location along the curve defined by the model:

ŷ _(k)={circumflex over (θ)}₁+{circumflex over (θ)}₂ u _(k)+{circumflexover (θ)}₃ u _(k) ²+∈_(k)

In some embodiments, the gradient is estimated using a recursive leastsquares (RLS) estimation technique with exponential forgetting. Any of avariety of RLS techniques can be used in various implementations. Anexample of a RLS technique which can be performed to estimate thegradient is shown in the following equations, which can be solved todetermine the value for the parameter vector {circumflex over (θ)}_(k).

g _(k) =P _(k-1) x _(k)(λ+x _(k) ^(T) P _(k-1) x _(k))⁻¹

P _(k)=λ⁻¹ P _(k-1) −g _(j) x _(k) ^(T)λ⁻¹ P _(k-1)

{circumflex over (θ)}_(k)={circumflex over (θ)}_(k-1) +e _(k) g _(k)

where g_(k) is a gain vector, P_(k) is a covariance matrix, and λ is aforgetting factor (λ<1). In some embodiments, the forgetting factor λ isdefined as follows:

$\lambda = e^{- \frac{\Delta \; t}{\tau}}$

where Δt is the sampling period and τ is the forgetting time constant.Once the parameter vector {circumflex over (θ)}_(k) is calculated, thegradient can be estimated by extracting the value of the {circumflexover (θ)}₂ parameter from {circumflex over (θ)}_(k)

In various embodiments, the gradient can be estimated using any of avariety of other recursive estimation techniques. For example, thegradient can be estimated using a Kalman filter, a normalized gradienttechnique, an unnormalized gradient adaption technique, a recursiveBayesian estimation technique, or any of a variety of linear ornonlinear filters. In some embodiments, the gradient can be estimatedusing a batch estimation technique rather than a recursive estimationtechnique. In the batch estimation technique, a batch of previous valuesfor the control input u and the performance variable y (e.g., a vectoror set of previous or historical values) can be used as inputs to abatch regression algorithm. Suitable regression algorithms may include,for example, ordinary least squares regression, polynomial regression,partial least squares regression, ridge regression, principal componentregression, or any of a variety of linear or nonlinear regressiontechniques.

In some embodiments, the gradient can be estimated using the covariancebetween the control input u and the performance variable y. For example,the estimate of the slope {circumflex over (β)} in a least-squaresapproach can be defined as:

$\hat{\beta} = \frac{{Cov}\left( {u,y} \right)}{{Var}(u)}$

where Cov(u,y) is the covariance between the control input u and theperformance variable y, and Var(u) is the variance of the control inputu. The estimated slope {circumflex over (β)} can be calculated using theprevious equation and used as a proxy for the gradient

$\frac{dy}{du}.$

Still referring to FIG. 8, flow diagram 800 is shown to include drivingthe estimated gradient toward zero by modulating an output of a feedbackcontroller (block 808). In some embodiments, the feedback controller isfeedback controller 508 shown in FIG. 5. The feedback controller canreceive the estimated gradient as an input and can modulate its output(e.g., DC output w) to drive the estimated gradient toward zero. Thefeedback controller can increase or decrease the value of the DC outputw until an optimum value for the DC output w is reached. The optimumvalue of the DC output w can be defined as the value which results in anoptimum value (e.g., a maximum or minimum value) of the performancevariable y. The optimum value of the performance variable y occurs whenthe gradient is zero. Accordingly, the feedback controller can achievethe optimum value of the performance variable y by modulating its outputw to drive the gradient to zero.

Flow diagram 800 is shown to include generating a stochastic excitationsignal q (block 810) and generating a new control input u by perturbingthe output w of the feedback controller with the stochastic excitationsignal q (block 812). The stochastic excitation signal q can begenerated by stochastic signal generator 512 and/or integrator 514, asdescribed with reference to FIG. 5. In various embodiments, thestochastic signal can be a random signal (e.g., a random walk signal, awhite noise signal, etc.), a non-periodic signal, an unpredictablesignal, a disturbance signal, or any other type of non-deterministic ornon-repeating signal. In some embodiments, the stochastic signal has anon-zero mean. The stochastic signal can be integrated to generate theexcitation signal q.

The stochastic excitation signal q can be added to the DC value wgenerated by the feedback controller to form the new control input u(e.g., u=w+q). After the new control input u is generated, the newcontrol input u can be provided to the plant (block 802) and the ESCcontrol technique can be repeated. The stochastic excitation signal qcan provide variation in the control input u sufficient to estimate theperformance gradient in block 806. In some instances, the addition ofexcitation signal q causes the control input u to drift away from itsoptimum value. However, the feedback controller can compensate for suchdrift by adjusting the DC value w such that the control input u iscontinuously pulled back toward its optimum value. As with traditionalESC, the magnitude of the excitation signal q can be selected (e.g.,manually by a user or automatically by the controller) to overcome anyadditive noise found in the performance variable y (e.g., process noise,measurement noise, etc.).

The stochastic excitation signal q has several advantages over aperiodic dither signal v. For example, the stochastic excitation signalq is less perceptible than a traditional periodic dither signal v. Assuch, the effects of the stochastic excitation signal q on the controlinput u are less noticeable than the periodic oscillations caused by thetraditional periodic dither signal v. Another advantage of thestochastic excitation signal q is that tuning the controller is simplerbecause the dither frequency ω_(v) is no longer a required parameter.Accordingly, the controller does not need to know or estimate thenatural frequency of the plant when generating the stochastic excitationsignal q.

Referring now to FIG. 9, a flow diagram 900 illustrating anotherextremum-seeking control (ESC) technique is shown, according to someembodiments. The ESC technique shown in flow diagram 900 can beperformed by one or more components of a feedback controller (e.g.,controller 502) to monitor and control a plant (e.g., plant 504). Forexample, controller 502 can use the ESC technique to estimate anormalized correlation coefficient ρ relating an output of the plant(e.g., performance variable y) to a control input u provided to theplant. Controller 502 can determine an optimal value of the controlinput u by driving the normalized correlation coefficient ρ to zero.

Flow diagram 900 is shown to include providing a control input u to aplant (block 902) and receiving a performance variable y as a feedbackfrom a plant (block 904). The control input u can be provided by anextremum-seeking controller and/or a feedback controller for the plant.The controller can be any of the controllers previously described (e.g.,controller 302, controller 402, controller 502, etc.) or any other typeof controller that provides a control input u to a plant. In someembodiments, the controller is an extremum-seeking controller configuredto achieve an optimal value for the performance variable y by adjustingthe control input u. The optimal value can be an extremum (e.g., amaximum or a minimum) of the performance variable y.

A plant in control theory is the combination of a process and one ormore mechanically-controlled outputs. The plant can be any of the plantspreviously described (e.g., plant 304, plant 404, plant 504, etc.) orany other controllable system or process. For example, the plant can bean air handling unit configured to control temperature within a buildingspace via one or more mechanically-controlled actuators and/or dampers.In various embodiments, the plant can include a chiller operationprocess, a damper adjustment process, a mechanical cooling process, aventilation process, a refrigeration process, or any other process inwhich a control input u to the plant is adjusted to affect theperformance variable y. The performance variable y can be a measuredvariable observed by one or more sensors of the plant (e.g., a measuredtemperature, a measured power consumption, a measured flow rate, etc.),a calculated variable based on measured or observed values (e.g., acalculated efficiency, a calculated power consumption, a calculatedcost, etc.) or any other type of variable that indicates the performanceof the plant in response to the control input u.

Flow diagram 900 is shown to include estimating a normalized correlationcoefficient ρ relating the performance variable y to the control inputu. The correlation coefficient ρ can be related to the performancegradient

$\frac{dy}{du}\mspace{11mu} \left( {{e.g.},{{proportional}\mspace{14mu} {to}\mspace{14mu} \frac{dy}{du}}} \right)$

but scaled based on the range of the performance variable y. Forexample, the correlation coefficient ρ can be a normalized measure ofthe performance gradient

$\frac{dy}{du}$

(e.g., scaled to the range 0≤ρ≤1).

In some embodiments, the correlation coefficient ρ can be estimatedbased on the variance and covariance of the control input u and theperformance variable y, as shown in the following equation:

$\rho = \frac{{Cov}\; \left( {u,y} \right)}{\sqrt{{Var}\; (u){{Var}(y)}}}$

where Cov(u,y) is the covariance between the control input u and theperformance variable y, Var(u) is the variance of the control input u,and Var(y) is the variance of the performance variable y. The previousequation can be rewritten in terms of the standard deviation σ_(u) ofthe control input u and the standard deviation σ_(y) of the performancevariable y as follows:

$\rho = \frac{{Cov}\left( {u,y} \right)}{\sigma_{u}\mspace{11mu} \sigma_{y}}$

where Var(u)=σ_(u) ² and Var(y)=σ_(y) ²

In some embodiments, the correlation coefficient ρ is estimated using arecursive estimation technique. The recursive estimation technique mayinclude calculating exponentially-weighted moving averages (EWMAs) ofthe control input u and the performance variable y. For example, EWMAsof the control input u and the performance variable y can be calculatedusing the following equations:

${\overset{\_}{u}}_{k} = {{\overset{\_}{u}}_{k - 1} + \frac{u_{k} - {\overset{\_}{u}}_{k - 1}}{\min \left( {k,W} \right)}}$${\overset{\_}{y}}_{k} = {{\overset{\_}{y}}_{k - 1} + \frac{y_{k} - {\overset{\_}{y}}_{k - 1}}{\min \left( {k,W} \right)}}$

where ū_(k) and y _(k) are the EWMAs of the control input u and theperformance variable y at time k, ū_(k-1) and y _(k-1) are the previousEWMAs of the control input u and the performance variable y at time k−1,u_(k) and y_(k) are the current values of the control input u and theperformance variable y at time k, k is the total number of samples thathave been collected of each variable, and W is the duration of theforgetting window.

EWMAs can also be calculated for the control input variance Var(u), theperformance variable variance Var(y), and the covariance Cov(u,y) usingthe following equations:

$\begin{matrix}{V_{u,k} = {V_{u,{k - 1}} + \frac{\left( {u_{k} - {\overset{\_}{u}}_{k}} \right)^{2} - V_{u,{k - 1}}}{\min \left( {k,W} \right)}}} \\{V_{y,k} = {V_{y,{k - 1}} + \frac{\left( {y_{k} - {\overset{\_}{y}}_{k}} \right)^{2} - V_{y,{k - 1}}}{\min \left( {k,W} \right)}}} \\{c_{k} = {c_{k - 1} + \frac{{\left( {y_{k} - {\overset{\_}{y}}_{k}} \right)\left( {u_{k} - {\overset{\_}{u}}_{k}} \right)} - c_{k - 1}}{\min \left( {k,W} \right)}}}\end{matrix}$

where V_(u,k), V_(y,k), and c_(k) are the EWMAs of the control inputvariance Var(u), the performance variable variance Var(y), and thecovariance Cov(u,y), respectively, at time k. V_(u,k-1), V_(y,k-1), andc_(k-1) are the EWMAs of the control input variance Var(u), theperformance variable variance Var(y), and the covariance Cov(u,y),respectively, at time k−1. The correlation coefficient ρ can beestimated based on these recursive estimates using the followingequation:

${\hat{\rho}}_{k} = \frac{c_{k}}{\sqrt{V_{u,k}V_{y,k}}}$

In some embodiments, the correlation coefficient ρ is estimated based onthe estimated slope {circumflex over (β)}. As previously described, theestimated slope {circumflex over (β)} can be calculated using thefollowing equation:

$\hat{\beta} = {\frac{{Cov}\left( {u,y} \right)}{{Var}\; (u)} = \frac{{Cov}\left( {u,y} \right)}{\sigma_{u}^{2}}}$

where Cov(u,y) is the covariance between the control input u and theperformance variable y, and Var(u) is the variance of the control inputu (i.e., σ_(u) ²). The correlation coefficient ρ can be calculated fromthe slope {circumflex over (β)} using the following equation:

$\rho = {\hat{\beta}\frac{\sigma_{u}}{\sigma_{y}}}$

From the previous equation, it can be seen that the correlationcoefficient ρ and the estimated slope {circumflex over (β)} are equalwhen the standard deviations σ_(u) and σ_(y) are equal (i.e., whenσ_(u)=σ_(y)).

In some embodiments, the estimated slope {circumflex over (β)} can becalculated using a set of values for the control input u and theperformance variable y. For example, with the assumption of finitevariance in u and y, the slope {circumflex over (β)} can be estimatedusing the following least squares estimation:

$\hat{\beta} = {\left( {\sum\limits_{i = {t - N}}^{t}{u_{i}u_{i}^{T}}} \right)^{- 1}\left( {\sum\limits_{i = {t - N}}^{t}{u_{i}y_{i}}} \right)}$

For a small range of the control input u, the estimated slope{circumflex over (β)} can be used as a proxy for the performancegradient, as shown in the following equation:

$\hat{\beta} = {\frac{dy}{du} = {{Kf}^{\mspace{11mu} \prime}(u)}}$

As shown in the previous equation, the estimated slope {circumflex over(β)} contains the constant gain K, which may be unknown. However,normalization provided by the standard deviations σ_(u) and σ_(y)cancels the effect of the constant gain K. For example, the standarddeviation σ_(y) of the performance variable y is related to the standarddeviation σ_(u) of the control input u as shown in the followingequations:

σ_(y) = K σ_(u) $\frac{\sigma_{u}}{\sigma_{y}} = \frac{1}{K}$

Multiplying the estimated slope {circumflex over (β)} by the ratio

$\frac{\sigma_{u}}{\sigma_{y}}$

to calculate the correlation coefficient ρ is equivalent to dividing bythe constant gain K. Both the correlation coefficient ρ and theestimated slope {circumflex over (β)} indicate the strength of therelationship between the control input u and the performance variable y.However, the correlation coefficient ρ has the advantage of beingnormalized which makes tuning the feedback control loop much simpler.

Still referring to FIG. 9, flow diagram 900 is shown to include drivingthe estimated correlation coefficient ρ toward zero by modulating anoutput of a feedback controller (block 908). In some embodiments, thefeedback controller is feedback controller 508 shown in FIG. 5. Thefeedback controller can receive the estimated correlation coefficient ρas an input and can modulate its output (e.g., DC output w) to drive theestimated correlation coefficient ρ toward zero. The feedback controllercan increase or decrease the value of the DC output w until an optimumvalue for the DC output w is reached. The optimum value of the DC outputw can be defined as the value which results in an optimum value (e.g., amaximum or minimum value) of the performance variable y. The optimumvalue of the performance variable y occurs when the gradient is zero.Accordingly, the feedback controller can achieve the optimum value ofthe performance variable y by modulating its output w to drive theestimated correlation coefficient ρ to zero.

Flow diagram 900 is shown to include generating an excitation signal(block 910) and generating a new control input u by perturbing theoutput w of the feedback controller with the excitation signal (block912). In various embodiments, the excitation signal can be a periodicdither signal v as described with reference to FIGS. 3-4 or a stochasticexcitation signal q as described with reference to FIG. 5. Theexcitation signal can be added to the DC value w generated by thefeedback controller to form the new control input u (e.g., u=w+q oru=w+v). After the new control input u is generated, the new controlinput u can be provided to the plant (block 902) and the ESC controltechnique can be repeated.

The excitation signal can provide variation in the control input usufficient to estimate the correlation coefficient ρ in block 906. Insome instances, the addition of the excitation signal causes the controlinput u to drift away from its optimum value. However, the feedbackcontroller can compensate for such drift by adjusting the DC value wsuch that the control input u is continuously pulled back toward itsoptimum value. The magnitude of the excitation signal can be selected(e.g., manually by a user or automatically by the controller) toovercome any additive noise found in the performance variable y (e.g.,process noise, measurement noise, etc.).

Example Implementations

Referring now to FIGS. 10A-16C several example implementations of theextremum seeking control systems and methods of the present disclosureare shown. The implementations shown in FIGS. 10A-16C illustrate variousembodiments of plant 504 which can be controlled by extremum seekingcontroller 502, the control input(s) u which can be provided to plant504 by extremum seeking controller 502, and the performance variable(s)y which can be received as feedback from plant 504 by extremum seekingcontroller 502.

Chilled Water Plant 1000

Referring particularly to FIG. 10A, a chilled water plant 1000 is shown,according to some embodiments. Chilled water plant 1000 is shown toinclude a chiller 1002, a cooling tower 1004, and an air handling unit(AHU) 1006. Chiller 1002 includes a condenser 1018, an evaporator 1020,and a compressor 1034. Compressor 1034 is configured to circulate arefrigerant between condenser 1018 and evaporator 1020 via a refrigerantloop 1026. Chiller 1002 also includes at least one expansion valve onrefrigerant loop 1026 between condenser 1018 and evaporator 1020.Chiller 1002 operates using a vapor compression refrigeration cycle inwhich the refrigerant in refrigerant loop 1026 absorbs heat inevaporator 1020 and rejects heat in condenser 1018. Chiller 1002 caninclude any number of sensors, control valves, and/or other componentsthat assist the refrigeration cycle operation of chiller 1002.

Chiller 1002 is connected with cooling tower 1004 by a condenser waterloop 1022. A water pump 1014 located along condenser water loop 1022circulates condenser water between cooling tower 1004 and chiller 1002via condenser water loop 1022. Pump 1014 can be a fixed speed pump or avariable speed pump. Condenser water loop 1022 circulates the condenserwater through condenser 1018 where the condenser water absorbs heat fromthe refrigerant in refrigeration loop 1026. The heated condenser wateris then delivered to cooling tower 1004 where the condenser waterrejects heat to the ambient environment. A cooling tower fan system 1036provides airflow through cooling tower 1004 to facilitate cooling thecondenser water within cooling tower 1004. The cooled condenser water isthen pumped back to chiller 1002 by pump 1014.

Chiller 1002 is connected with AHU 1006 via a chilled fluid loop 1024. Achilled fluid pump 1016 located along chilled fluid loop 1024 circulatesa chilled fluid between chiller 1002 and AHU 1006. Pump 1016 can be afixed speed pump or a variable speed pump. Chilled fluid loop 1024circulates the chilled fluid through evaporator 1020 where the chilledfluid rejects heat to the refrigerant in refrigeration loop 1026. Thechilled fluid is then delivered to AHU 1006 where the chilled fluidabsorbs heat from the supply air passing through AHU 1006, therebyproviding cooling for the supply air. The heated fluid is then pumpedback to chiller 1002 by pump 1016.

In the embodiment shown in FIG. 10A, AHU 1006 is shown as an economizertype air handling unit. Economizer type AHUs vary the amount of outdoorair and return air used by the AHU for cooling. AHU 1006 is shown toinclude economizer controller 1032 that utilizes one or more algorithms(e.g., state based algorithms, extremum seeking control algorithms,etc.) to affect the actuators and dampers or fans of AHU 1006. The flowof chilled fluid supplied to AHU 1006 can also be variably controlled.For example, PI control 1008 is shown controlling a valve 1038 thatregulates the flow of the chilled fluid to AHU 1006. PI control 1008 cancontrol the chilled fluid flow to AHU 1006 to achieve a setpoint supplyair temperature. Economizer controller 1032, a controller for chiller1002, and PI control 1008 can be supervised by one or more buildingmanagement system (BMS) controllers 1010.

A BMS controller is, in general, a computer-based system configured tocontrol, monitor, and manage equipment in or around a building orbuilding area. A BMS controller can include a METASYS® brand buildingcontroller or other devices sold by Johnson Controls, Inc. BMScontroller 1010 can provide one or more human-machine interfaces orclient interfaces (e.g., graphical user interfaces, reportinginterfaces, text-based computer interfaces, client-facing web services,web servers that provide pages to web clients, etc.) for controlling,viewing, or otherwise interacting with the BMS, its subsystems, anddevices. For example, BMS controller 1010 can provide a web-basedgraphical user interface that allows a user to set a desired setpointtemperature for a building space. BMS controller 1010 can use BMSsensors 1012 (connected to BMS controller 1010 via a wired or wirelessBMS or IT network) to determine if the setpoint temperatures for thebuilding space are being achieved. BMS controller 1010 can use suchdeterminations to provide commands to PI control 1008, chiller 1002,economizer controller 1032, or other components of the building's HVACsystem.

In some embodiments, extremum seeking controller 502 does not receivecontrol commands from BMS controller 1010 or does not base its outputcalculations on an input from BMS controller 1010. In other embodiments,extremum seeking controller 502 receives information (e.g., commands,setpoints, operating boundaries, etc.) from BMS controller 1010. Forexample, BMS controller 1010 can provide extremum seeking controller 502with a high fan speed limit and a low fan speed limit. A low limit mayavoid frequent component and power taxing fan start-ups while a highlimit can avoid operation near the mechanical or thermal limits of thefan system.

Extremum seeking controller 502 is shown receiving a power inputP_(total) representing the total power consumed by cooling tower fansystem 1036 P_(tower), condenser water pump 1014 P_(pump), and thecompressor 1034 of chiller 1002 P_(chiller) (i.e.P_(total)=P_(tower)+P_(pump)+P_(chiller)) As illustrated in FIG. 10A,the power inputs P_(tower), P_(pump), and P_(chiller) can be summedoutside of extremum seeking controller 502 at summing block 1040 toprovide a combined signal representative of the total power P_(total.)In other embodiments, extremum seeking controller 502 receives theindividual power inputs P_(tower), P_(pump), and P_(chiller) andconducts the summation of summing block 1040. In either case, extremumseeking controller 502 can be said to receive the power inputsP_(tower), P_(pump), and P_(chiller) even if the power inputs areprovided as a single summed or combined signal P_(total) representingthe total system power.

In some embodiments, the total system power P_(total) is the performancevariable which extremum seeking controller 502 seeks to optimize (e.g.,minimize). The total system power P_(total) can include the powerconsumption of one or more components of chilled water plant 1000. Inthe embodiment shown in FIG. 10A, the total system power P_(total)includes P_(tower), P_(pump), and P_(chiller). However, in various otherembodiments, the total system power P_(total) can include anycombination of power inputs. For example, the total system powerP_(total) can include the power consumption of the fans within AHU 1006,the power consumption of chilled fluid pump 1016, and/or any other powerconsumption that occurs within chilled water plant 1000.

Extremum seeking controller 502 is shown providing a temperaturesetpoint T_(sp) to a feedback controller 1028. In some embodiments, thetemperature setpoint T_(sp) is the manipulated variable which extremumseeking controller 502 adjusts to affect the total system powerP_(total). The temperature setpoint T_(sp) is a setpoint for thetemperature of the condenser water T_(cw) provided to chiller 1002 fromcooling tower 1004. The condenser water temperature T_(cw) can bemeasured by a temperature sensor 1030 located along condenser water loop1022 between cooling tower 1004 and chiller 1002 (e.g., upstream ordownstream of pump 1014). Feedback controller 1028 is shown receivingthe condenser water temperature T_(cw) as a feedback signal.

Feedback controller 1028 can operate cooling tower fan system 1036and/or condenser water pump 1014 to achieve the temperature setpointT_(sp) provided by extremum seeking controller 502. For example,feedback controller 1028 can increase the speed of condenser water pump1014 to increase the amount of heat removed from the refrigerant incondenser 1018 or decrease the speed of condenser water pump 1014 todecrease the amount of heat removed from the refrigerant in condenser1018. Similarly, feedback controller 1028 can increase the speed ofcooling tower fan system 1036 to increase the amount of heat removedfrom the condenser water by cooling tower 1004 or decrease the speed ofcooling tower fan system 1036 to decrease the amount of heat removedfrom the condenser water by cooling tower 1004.

Extremum seeking controller 502 implements an extremum seeking controlstrategy that dynamically searches for an unknown input (e.g., optimalcondenser water temperature setpoint T_(sp)) to obtain systemperformance (e.g., total power consumption P_(total)) that trends nearoptimal. Although feedback controller 1028 and extremum seekingcontroller 502 are shown as separate devices, it is contemplated thatfeedback controller 1028 and extremum seeking controller 502 can becombined into a single device in some embodiments (e.g., a singlecontroller that performs the functions of both extremum seekingcontroller 502 and feedback controller 1028). For example, extremumseeking controller 502 can be configured to control cooling tower fansystem 1036 and condenser water pump 1014 directly without requiring anintermediate feedback controller 1028.

Referring now to FIGS. 10B and 10C, a pair of flow diagrams 1050 and1070 illustrating the operation of extremum controller 502 in chilledwater plant 1000 are shown, according to some embodiments. In both flowdiagrams 1050 and 1070, extremum seeking controller 502 provides atemperature setpoint T_(sp) to a feedback controller 1028 that operatesto control condenser water temperature T_(cw) in a chilled water plant1000 (blocks 1052 and 1072). Extremum seeking controller 502 can receivea total power consumption P_(total) of the chilled water plant 1000 as afeedback signal (blocks 1054 and 1074).

In flow diagram 1050, extremum seeking controller 502 estimates agradient of the total power consumption P_(total) with respect to thecondenser water temperature setpoint T_(sp) (block 1056). Extremumseeking controller 502 can provide control over the chilled water plant1000 by driving the obtained gradient toward zero by modulating thetemperature setpoint T_(sp) (block 1058). In some embodiments, extremumseeking controller 502 generates a stochastic excitation signal (block1060) and uses the stochastic excitation signal to generate a newcondenser water temperature setpoint T_(sp). For example, extremumseeking controller 502 can generate the new temperature setpoint T_(sp)by perturbing the condenser water temperature setpoint T_(sp) with thestochastic excitation signal (block 1062).

In flow diagram 1070, extremum seeking controller 502 estimates anormalized correlation coefficient relating the total power consumptionP_(total) to the condenser water temperature setpoint T_(sp) (block1076). Extremum seeking controller 502 can provide control over thechilled water plant 1000 by driving the estimated correlationcoefficient toward zero by modulating the temperature setpoint T_(sp)(block 1078). In some embodiments, extremum seeking controller 502generates an excitation signal (block 1080) and uses the excitationsignal to generate a new condenser water temperature setpoint T_(sp).For example, extremum seeking controller 502 can generate the newtemperature setpoint T_(sp) by perturbing the condenser watertemperature setpoint T_(sp) with the excitation signal (block 1082).

Chilled Water Plant 1100

Referring now to FIG. 11A, another chilled water plant 1100 is shown,according to some embodiments. Chilled water plant 1100 can include someor all of the components of chilled water plant 1000, as described withreference to FIG. 10A. For example, chilled water plant 1100 is shown toinclude a chiller 1102, a cooling tower 1104, and an air handling unit(AHU) 1106. Chiller 1102 is connected with cooling tower 1104 by acondenser water loop 1122. A water pump 1114 located along condenserwater loop 1122 circulates condenser water between cooling tower 1104and chiller 1102. A cooling tower fan system 1136 provides airflowthrough cooling tower 1104 to facilitate cooling the condenser waterwithin cooling tower 1104. Chiller 1002 is also connected with AHU 1106via a chilled fluid loop 1124. A chilled fluid pump 1116 located alongchilled fluid loop 1124 circulates a chilled fluid between chiller 1102and AHU 1106.

Extremum seeking controller 502 is shown receiving a power input P totalrepresenting the total power consumed by cooling tower fan system 1136P_(tower), condenser water pump 1114 P_(pump), and the compressor 1134of chiller 1102 P_(chiller) (i.e.P_(total)=P_(tower)+P_(pump)+P_(chiller)). In some embodiments, thetotal system power P_(total) is the performance variable which extremumseeking controller 502 seeks to optimize (e.g., minimize). In theembodiment shown in FIG. 11A, the total system power P_(total) includesP_(tower), P_(pump), and P_(chiller). However, in various otherembodiments, the total system power P_(total) can include anycombination of power inputs. For example, the total system powerP_(total) can include the power consumption of the fans within AHU 1106,the power consumption of chilled fluid pump 1116, and/or any other powerconsumption that occurs within chilled water plant 1100.

Extremum seeking controller 502 is shown providing a first controlsignal regulating the fan speed Fan_(sp) of cooling tower fan system1136 and a second control signal regulating the pump speed Pump_(sp) ofcondenser water pump 1114. In some embodiments, the fan speed Fan_(sp)and the pump speed Pump_(sp) are the manipulated variables whichextremum seeking controller 502 adjusts to affect the total system powerP_(total.) For example, extremum seeking controller 502 can increase thepump speed Pump_(sp) to increase the amount of heat removed from therefrigerant in condenser 1118 or decrease the pump speed Pump_(sp) todecrease the amount of heat removed from the refrigerant in condenser1118. Similarly, extremum seeking controller 502 can increase the fanspeed Fan_(sp) to increase the amount of heat removed from the condenserwater by cooling tower 1104 or decrease the fan speed Fan_(sp) todecrease the amount of heat removed from the condenser water by coolingtower 1104.

Referring now to FIGS. 11B and 11C, a pair of flow diagrams 1150 and1170 illustrating the operation of extremum controller 502 in chilledwater plant 1100 are shown, according to some embodiments. In both flowdiagrams 1150 and 1170, extremum seeking controller 502 provides a fanspeed control signal Fan_(sp) to a cooling tower fan system and a pumpspeed control signal Pump_(sp) to a condenser water pump (blocks 1152and 1172). Extremum seeking controller 502 can receive a total powerconsumption P_(total) of the chilled water plant 1100 as a feedbacksignal (blocks 1154 and 1174).

In flow diagram 1150, extremum seeking controller 502 estimates a firstgradient of the total power consumption P_(total) with respect to thefan speed Fan_(sp) and a second gradient of the total power consumptionP_(total) with respect to the condenser water pump speed Pump_(sp)(block 1156). Extremum seeking controller 502 can provide control overthe chilled water plant 1100 by driving the obtained gradients towardzero by modulating the fan speed Fan_(sp) and the condenser water pumpspeed Pump_(sp) (block 1158). In some embodiments, extremum seekingcontroller 502 generates a stochastic excitation signal for each of thespeed control signals (block 1160) and uses the stochastic excitationsignals to generate a new speed control signals (block 1162). Forexample, extremum seeking controller 502 can generate a new fan speedcontrol signal Fan_(sp) by perturbing the fan speed control signalFan_(sp) with a first stochastic excitation signal. Extremum seekingcontroller 502 can generate a new pump speed control signal Pump_(sp) byperturbing the pump speed control signal Pump_(sp) with a secondstochastic excitation signal.

In flow diagram 1070, extremum seeking controller 502 estimates a firstnormalized correlation coefficient relating the total power consumptionP total to the fan speed Fan_(sp) and a second normalized correlationcoefficient relating the total power consumption P total to thecondenser water pump speed Pump_(sp) (block 1176). Extremum seekingcontroller 502 can provide control over the chilled water plant 1100 bydriving the estimated correlation coefficients toward zero by modulatingthe fan speed Fan_(sp) and the pump speed Pump_(sp) (block 1178). Insome embodiments, extremum seeking controller 502 generates anexcitation signal for each of the speed control signals (block 1080) anduses the excitation signals to generate new fan and pump speeds (block1182). For example, extremum seeking controller 502 can generate a newfan speed control signal Fan_(sp) by perturbing the fan speed controlsignal Fan_(sp) with a first excitation signal. Extremum seekingcontroller 502 can generate a new pump speed control signal Pump_(sp) byperturbing the pump speed control signal Pump_(sp) with a secondexcitation signal.

Variable Refrigerant Flow System 1200

Referring now to FIG. 12A, a variable refrigerant flow (VRF) system 1200is shown, according to some embodiments. VRF system 1200 is shown toinclude an outdoor unit 1202, several heat recovery units 1204, andseveral indoor units 1206. In some embodiments, outdoor unit 1202 islocated outside a building (e.g., on a rooftop) whereas indoor units1206 are distributed throughout the building (e.g., in various rooms orzones of the building). In some embodiments, VRF system 1200 includesseveral heat recovery units 1204. Heat recovery units 1204 can controlthe flow of a refrigerant between outdoor unit 1204 and indoor units1206 (e.g., by opening or closing valves) and can minimize the heatingor cooling load to be served by outdoor unit 1202.

Outdoor unit 1202 is shown to include a compressor 1214 and a heatexchanger 1220. Compressor 1214 circulates a refrigerant between heatexchanger 1220 and indoor units 1206. Heat exchanger 1220 can functionas a condenser (allowing the refrigerant to reject heat to the outsideair) when VRF system 1200 operates in a cooling mode or as an evaporator(allowing the refrigerant to absorb heat from the outside air) when VRFsystem 1200 operates in a heating mode. A fan 1218 provides airflowthrough heat exchanger 1220. The speed of fan 1218 can be adjusted tomodulate the rate of heat transfer into or out of the refrigerant inheat exchanger 1220.

Each indoor unit 1206 is shown to include a heat exchanger 1226 and anexpansion valve 1224. Each of heat exchangers 1226 can function as acondenser (allowing the refrigerant to reject heat to the air within theroom or zone) when the indoor unit 1206 operates in a heating mode or asan evaporator (allowing the refrigerant to absorb heat from the airwithin the room or zone) when the indoor unit 1206 operates in a coolingmode. Fans 1222 provide airflow through heat exchangers 1226. The speedsof fans 1222 can be adjusted to modulate the rate of heat transfer intoor out of the refrigerant in heat exchangers 1226. Temperature sensors1228 can be used to measure the temperature of the refrigerant withinindoor units 1206.

In FIG. 12A, indoor units 1206 are shown operating in the cooling mode.In the cooling mode, the refrigerant is provided to indoor units 1206via cooling line 1212. The refrigerant is expanded by expansion valves1224 to a cold, low pressure state and flows through heat exchangers1226 (functioning as evaporators) to absorb heat from the room or zonewithin the building. The heated refrigerant then flows back to outdoorunit 1202 via return line 1210 and is compressed by compressor 1214 to ahot, high pressure state. The compressed refrigerant flows through heatexchanger 1220 (functioning as a condenser) and rejects heat to theoutside air. The cooled refrigerant can then be provided back to indoorunits 1206 via cooling line 1212. In the cooling mode, flow controlvalves 1236 can be closed and expansion valve 1234 can be completelyopen.

In the heating mode, the refrigerant is provided to indoor units 1206 ina hot state via heating line 1208. The hot refrigerant flows throughheat exchangers 1226 (functioning as condensers) and rejects heat to theair within the room or zone of the building. The refrigerant then flowsback to outdoor unit via cooling line 1212 (opposite the flow directionshown in FIG. 12A). The refrigerant can be expanded by expansion valve1234 to a colder, lower pressure state. The expanded refrigerant flowsthrough heat exchanger 1220 (functioning as an evaporator) and absorbsheat from the outside air. The heated refrigerant can be compressed bycompressor 1214 and provided back to indoor units 1206 via heating line1208 in a hot, compressed state. In the heating mode, flow controlvalves 1236 can be completely open to allow the refrigerant fromcompressor 1214 to flow into heating line 1208.

Extremum seeking controller 502 is shown receiving a power inputP_(total) representing the total power consumed by outdoor unit 1202P_(outdoor) and the total power consumed by each of indoor units 1206P_(indoor) (i.e., P_(total)=P_(outdoor)+P_(indoor)). The outdoor unitpower outdoor can include the power consumption of compressor 1214and/or fan 1218. The indoor unit power P_(indoor) can include the powerconsumption of fans 1222 and/or any other power-consuming devices withinindoor units 1206 or heat recovery units 1204 (e.g., electronic valves,pumps, fans, etc.). As illustrated in FIG. 12A, the power inputsP_(outdoor) and P_(indoor) can be summed outside of extremum seekingcontroller 502 at summing block 1230 to provide a combined signalrepresentative of the total power P_(total). In other embodiments,extremum seeking controller 502 receives the individual power inputsP_(outdoor) and P_(indoor) and conducts the summation of summing block1230. In either case, extremum seeking controller 502 can be said toreceive the power inputs P_(outdoor) and P_(indoor) even if the powerinputs are provided as a single summed or combined signal P_(total)representing the total system power.

In some embodiments, the total system power P_(total) is the performancevariable which extremum seeking controller 502 seeks to optimize (e.g.,minimize). The total system power P_(total) can include the powerconsumption of one or more components of VRF system 1200. In theembodiment shown in FIG. 12A, the total system power P_(total) includesP_(outdoor) and P_(indoor). However, in various other embodiments, thetotal system power P_(total) can include any combination of powerinputs. For example, the total system power P_(total) can include thepower consumption of heat recovery units 1204, indoor units 1206,outdoor unit 1202, pumps, and/or any other power consumption that occurswithin VRF system 1200.

Extremum seeking controller 502 is shown providing a pressure setpointP_(sp) to an outdoor unit controller 1232. In some embodiments, thepressure setpoint P_(sp) is the manipulated variable which extremumseeking controller 502 adjusts to affect the total system powerP_(total). The pressure setpoint P_(sp) is a setpoint for the pressureof the refrigerant P_(r) at the suction or the discharge of compressor1214. The refrigerant pressure P_(r) can be measured by a pressuresensor 1216 located at the suction of compressor 1214 (e.g., upstream ofcompressor 1214) or at the discharge of compressor 1214 (e.g.,downstream of compressor 1214). Outdoor unit controller 1232 is shownreceiving the refrigerant pressure P_(r) as a feedback signal.

Outdoor unit controller 1232 can operate outdoor unit 1202 to achievethe pressure setpoint P_(sp) provided by extremum seeking controller502. Operating outdoor unit 1202 can include adjusting the speed ofcompressor 1214 and/or the speed of fan 1218. For example, outdoor unitcontroller 1232 can increase the speed of compressor 1214 to increasecompressor discharge pressure or decrease the compressor suctionpressure. Outdoor unit controller 1232 can increase the speed of fan1218 to increase the heat transfer within heat exchanger 1220 ordecrease the speed of fan 1218 to decrease the heat transfer within heatexchanger 1220.

Extremum seeking controller 502 implements an extremum seeking controlstrategy that dynamically searches for an unknown input (e.g., pressuresetpoint P_(sp)) to obtain system performance (e.g., total powerconsumption P_(total)) that trends near optimal. Although outdoor unitcontroller 1232 and extremum seeking controller 502 are shown asseparate devices, it is contemplated that outdoor unit controller 1232and extremum seeking controller 502 can be combined into a single devicein some embodiments (e.g., a single controller that performs thefunctions of both extremum seeking controller 502 and outdoor unitcontroller 1232). For example, extremum seeking controller 502 can beconfigured to operate compressor 1214 and/or fan 1218 directly withoutrequiring an intermediate outdoor unit controller 1232.

Referring now to FIGS. 12B and 12C, a pair of flow diagrams 1250 and1270 illustrating the operation of extremum controller 502 in VRF system1200 are shown, according to some embodiments. In both flow diagrams1250 and 1270, extremum seeking controller 502 provides a pressuresetpoint P_(sp) to a controller (e.g., outdoor unit controller 1232)that operates to control refrigerant pressure in an outdoor unit 1202 ofa VRF system 1200 (blocks 1252 and 1272). The refrigerant pressure canbe a compressor suction pressure or a compressor discharge pressure.Extremum seeking controller 502 can receive a total power consumption Ptotal of the VRF system 1200 as a feedback signal (blocks 1254 and1274).

In flow diagram 1250, extremum seeking controller 502 estimates agradient of the total power consumption P_(total) with respect to therefrigerant pressure setpoint P_(sp) (block 1256). Extremum seekingcontroller 502 can provide control over the VRF system 1200 by drivingthe obtained gradient toward zero by modulating the pressure setpointP_(sp) (block 1258). In some embodiments, extremum seeking controller502 generates a stochastic excitation signal (block 1260) and uses thestochastic excitation signal to generate a new refrigerant pressuresetpoint P_(sp). For example, extremum seeking controller 502 cangenerate the new pressure setpoint P_(sp) by perturbing the refrigerantpressure setpoint P_(sp) with the stochastic excitation signal (block1262).

In flow diagram 1270, extremum seeking controller 502 estimates anormalized correlation coefficient relating the total power consumptionP_(total) to the refrigerant pressure setpoint P_(sp) (block 1276).Extremum seeking controller 502 can provide control over the VRF system1200 by driving the estimated correlation coefficient toward zero bymodulating the refrigerant pressure setpoint P_(sp) (block 1278). Insome embodiments, extremum seeking controller 502 generates anexcitation signal (block 1280) and uses the excitation signal togenerate a new refrigerant pressure setpoint P_(sp). For example,extremum seeking controller 502 can generate the new pressure setpointP_(sp) by perturbing the refrigerant pressure setpoint P_(sp) with theexcitation signal (block 1282).

Variable Refrigerant Flow System 1300

Referring now to FIG. 13A, another variable refrigerant flow (VRF)system 1300 is shown, according to some embodiments. VRF system 1300 caninclude some or all of the components of VRF system 1200, as describedwith reference to FIG. 12A. For example, VRF system 1300 is shown toinclude an outdoor unit 1302, several heat recovery units 1304, andseveral indoor units 1306.

Outdoor unit 1302 is shown to include a compressor 1314 and a heatexchanger 1320. Compressor 1314 circulates a refrigerant between heatexchanger 1320 and indoor units 1306. Heat exchanger 1320 can functionas a condenser (allowing the refrigerant to reject heat to the outsideair) when VRF system 1300 operates in a cooling mode or as an evaporator(allowing the refrigerant to absorb heat from the outside air) when VRFsystem 1300 operates in a heating mode. A fan 1318 provides airflowthrough heat exchanger 1220. The speed of fan 1318 can be adjusted tomodulate the rate of heat transfer into or out of the refrigerant inheat exchanger 1320.

Each indoor unit 1306 is shown to include a heat exchanger 1326 and anexpansion valve 1324. Each of heat exchangers 1326 can function as acondenser (allowing the refrigerant to reject heat to the air within theroom or zone) when the indoor unit 1306 operates in a heating mode or asan evaporator (allowing the refrigerant to absorb heat from the airwithin the room or zone) when the indoor unit 1306 operates in a coolingmode. Fans 1322 provide airflow through heat exchangers 1326. The speedsof fans 1322 can be adjusted to modulate the rate of heat transfer intoor out of the refrigerant in heat exchangers 1326. Temperature sensors1328 can be used to measure the temperature of the refrigerant T_(r)within indoor units 1306.

Extremum seeking controller 502 is shown receiving a power inputP_(total) representing the total power consumed by outdoor unit 1302P_(outdoor) and the total power consumed by each of indoor units 1306P_(indoor) (i.e., P_(total)=P_(outdoor)+P_(indoor)). The outdoor unitpower P_(outdoor) can include the power consumption of compressor 1314and/or fan 1318. The indoor unit power P_(indoor) can include the powerconsumption of fans 1322 and/or any other power-consuming devices withinindoor units 1306 or heat recovery units 1304 (e.g., electronic valves,pumps, fans, etc.).

In some embodiments, the total system power P_(total) is the performancevariable which extremum seeking controller 502 seeks to optimize (e.g.,minimize). The total system power P_(total) can include the powerconsumption of one or more components of VRF system 1300. In theembodiment shown in FIG. 13A, the total system power P_(total) includesP_(outdoor) and P_(indoor). However, in various other embodiments, thetotal system power P_(total) can include any combination of powerinputs. For example, the total system power P_(total) can include thepower consumption of heat recovery units 1304, indoor units 1306,outdoor unit 1302, pumps, and/or any other power consumption that occurswithin VRF system 1300.

Extremum seeking controller 502 is shown providing a pressure setpointP_(sp) to an outdoor unit controller 1332 and a superheat temperaturesetpoint T_(sp) to an indoor unit controller 1338. In some embodiments,the pressure setpoint P_(sp) and the superheat temperature setpointT_(sp) are the manipulated variables which extremum seeking controller502 adjusts to affect the total system power P_(total). The pressuresetpoint P_(sp) is a setpoint for the pressure of the refrigerant P_(r)at the suction or the discharge of compressor 1314. The superheattemperature setpoint T_(sp) is a setpoint for the amount of superheat ofthe refrigerant (i.e., the temperature of the refrigerant T_(r) minusthe refrigerant saturation temperature) at the outlet of heat exchangers1326.

The refrigerant pressure P_(r) can be measured by a pressure sensor 1316located at the suction of compressor 1314 (e.g., upstream of compressor1314) or at the discharge of compressor 1314 (e.g., downstream ofcompressor 1314). Outdoor unit controller 1332 is shown receiving therefrigerant pressure P_(r) as a feedback signal. Outdoor unit controller1232 can operate outdoor unit 1202 to achieve the pressure setpointP_(sp) provided by extremum seeking controller 502. Operating outdoorunit 1202 can include adjusting the speed of compressor 1214 and/or thespeed of fan 1218. For example, outdoor unit controller 1232 canincrease the speed of compressor 1214 to increase compressor dischargepressure or decrease the compressor suction pressure. Outdoor unitcontroller 1232 can increase the speed of fan 1218 to increase the heattransfer within heat exchanger 1220 or decrease the speed of fan 1218 todecrease the heat transfer within heat exchanger 1220.

The superheat of the refrigerant T_(super) can be calculated (by indoorunit controller 1338) by subtracting the refrigerant saturationtemperature T_(sat) from the temperature of the refrigerant T_(r) (i.e.,T_(super)=T_(r)−T_(sat)). The refrigerant temperature T_(r) can bemeasured by temperature sensors 1328 located at the outlet of heatexchangers 1326. Indoor unit controller 1338 is shown receiving therefrigerant pressure T_(r) as a feedback signal. Indoor unit controller1338 can operate indoor units 1306 to achieve the superheat temperaturesetpoint T_(sp) provided by extremum seeking controller 502. Operatingindoor units 1306 can include adjusting the speed of fans 1322 and/oradjusting the position of expansion valves 1324. For example, indoorunit controller 1338 can increase the speed of fans 1322 to increase theheat transfer within heat exchangers 1226 or decrease the speed of fans1322 to decrease the heat transfer within heat exchangers 1226.Similarly, indoor unit controller 1338 can move valves 1324 toward anopen position to increase the refrigerant flow through indoor units 1306or move valves 1324 toward a closed position to decrease the refrigerantflow through indoor units 1306.

Extremum seeking controller 502 implements an extremum seeking controlstrategy that dynamically searches for an unknown input (e.g., pressuresetpoint P_(sp) and/or superheat temperature setpoint T_(sp)) to obtainsystem performance (e.g., total power consumption P_(total)) that trendsnear optimal. Although outdoor unit controller 1332, indoor unitcontroller 1338, and extremum seeking controller 502 are shown asseparate devices, it is contemplated that outdoor unit controller 1332,indoor unit controller 1338, and extremum seeking controller 502 can becombined into a single device in some embodiments (e.g., a singlecontroller that performs the functions of extremum seeking controller502, outdoor unit controller 1332, and indoor unit controller 1338). Forexample, extremum seeking controller 502 can be configured to operatecompressor 1314, fan 1318, fans 1322, and/or valves 1224 directlywithout requiring an intermediate outdoor unit controller 1332 or indoorunit controller 1338.

Referring now to FIGS. 13B and 13C, a pair of flow diagrams 1350 and1370 illustrating the operation of extremum controller 502 in VRF system1300 are shown, according to some embodiments. In both flow diagrams1350 and 1370, extremum seeking controller 502 provides a pressuresetpoint P_(sp) to a controller (e.g., outdoor unit controller 1332)that operates to control refrigerant pressure in an outdoor unit 1302 ofa VRF system 1300 (blocks 1352 and 1372). The refrigerant pressure canbe a compressor suction pressure or a compressor discharge pressure.Extremum seeking controller 502 also provides a superheat temperaturesetpoint to a controller (e.g., indoor unit controller 1338) thatoperates to control refrigerant temperature in an indoor unit of the VRFsystem 1300 (blocks 1353 and 1373). Extremum seeking controller 502 canreceive a total power consumption P_(total) of the VRF system 1300 as afeedback signal (blocks 1354 and 1374).

In flow diagram 1350, extremum seeking controller 502 estimates a firstgradient of the total power consumption P_(total) with respect to therefrigerant pressure setpoint P_(sp) and a second gradient of the totalpower consumption P_(total) with respect to the refrigerant superheattemperature setpoint T_(sp) (block 1356). Extremum seeking controller502 can provide control over the VRF system 1300 by driving the obtainedgradients toward zero by modulating the pressure setpoint P_(sp) and thesuperheat temperature setpoint T_(sp) (block 1358). In some embodiments,extremum seeking controller 502 generates stochastic excitation signals(block 1360) and uses the stochastic excitation signals to generate anew refrigerant pressure setpoint P_(sp) and a new refrigerant superheatsetpoint T_(sp). For example, extremum seeking controller 502 cangenerate the new pressure setpoint P_(sp) by perturbing the refrigerantpressure setpoint P_(sp) with a first stochastic excitation signal andcan generate the new superheat temperature setpoint T_(sp) by perturbingthe temperature setpoint T_(sp) with a second stochastic excitationsignal (block 1362).

In flow diagram 1370, extremum seeking controller 502 estimates a firstnormalized correlation coefficient relating the total power consumptionP_(total) to the refrigerant pressure setpoint P_(sp) and a secondnormalized correlation coefficient relating the total power consumptionP_(total) to the refrigerant superheat temperature setpoint T_(sp)(block 1376). Extremum seeking controller 502 can provide control overthe VRF system 1300 by driving the estimated correlation coefficientstoward zero by modulating the refrigerant pressure setpoint P_(sp) andthe refrigerant superheat temperature setpoint T_(sp) (block 1378). Insome embodiments, extremum seeking controller 502 generates excitationsignals (block 1380) and uses the excitation signals to generate a newrefrigerant pressure setpoint P_(sp) and a new refrigerant superheatsetpoint T_(sp). For example, extremum seeking controller 502 cangenerate the new pressure setpoint P_(sp) by perturbing the refrigerantpressure setpoint P_(sp) with a first excitation signal and can generatethe new superheat temperature setpoint T_(sp) by perturbing thetemperature setpoint T_(sp) with a second excitation signal (block1382).

Vapor Compression System 1400

Referring now to FIG. 14A, a vapor compression air conditioning system1400 is shown, according to some embodiments. System 1400 is shown toinclude a refrigerant circuit 1410. Refrigerant circuit 1410 includes acondenser 1412, an evaporator 1414, an expansion valve 1424, and acompressor 1406. Compressor 1406 is configured to circulate arefrigerant between evaporator 1414 and condenser 1412. Refrigerantcircuit 1410 operates using a vapor compression cycle. For example,compressor 1406 compresses the refrigerant to a hot, high pressurestate. The compressed refrigerant flows through condenser 1412 where therefrigerant rejects heat. A condenser fan 1422 can be used to modulatethe rate of heat transfer within condenser 1412. The cooled refrigerantis expanded by expansion valve 1424 to a low pressure, low temperaturestate. The expanded refrigerant flows through evaporator 1414 where therefrigerant absorbs heat. An evaporator fan 1416 can be used to modulatethe rate of heat transfer within evaporator 1414.

In some embodiments, refrigerant circuit 1410 is located within arooftop unit 1402 (e.g., a rooftop air handling unit) as shown in FIG.14A. Rooftop unit 1402 can be configured to provide cooling for supplyair 1420 flowing through an air duct 1422. For example, evaporator 1414can be located within air duct 1422 such that supply air 1420 flowsthrough evaporator 1414 and is cooled by transferring heat to theexpanded refrigerant within evaporator 1414. The cooled airflow can thenbe routed to a building to provide cooling for a room or zone of thebuilding. The temperature of supply air 1420 can be measured by atemperature sensor 1418 located downstream of evaporator 1414 (e.g.,within duct 1422). In other embodiments, refrigerant circuit 1410 can beused in any of a variety of other systems or devices that transfer heatusing a vapor compression cycle (e.g., chillers, heat pumps, heatrecovery chillers, refrigeration devices, etc.).

Extremum seeking controller 502 is shown receiving a power inputP_(total) representing the total power consumed by compressor 1406P_(comp), evaporator fan 1416 P_(fan,evap), and condenser fan 1422P_(fan,cond) (i.e., P_(total)=P_(comp)+P_(fan,evap)+P_(fan,cond)). Asillustrated in FIG. 14A, the power inputs P_(comp), P_(fan,evap), andP_(fan,cond) can be summed outside of extremum seeking controller 502 atsumming block 1408 to provide a combined signal representative of thetotal power P_(total). In other embodiments, extremum seeking controller502 receives the individual power inputs P_(comp), P_(fan,evap), andP_(fan,cond) and conducts the summation of summing block 1408. In eithercase, extremum seeking controller 502 can be said to receive the powerinputs P_(comp), P_(fan,evap), and P_(fan,cond) even if the power inputsare provided as a single summed or combined signal P_(total)representing the total system power.

In some embodiments, the total system power P_(total) is the performancevariable which extremum seeking controller 502 seeks to optimize (e.g.,minimize). The total system power P_(total) can include the powerconsumption of one or more components of vapor compression system 1400.In the embodiment shown in FIG. 14A, the total system power P_(total)includes P_(comp), P_(fan,evap), and P_(fan,cond). However, in variousother embodiments, the total system power P_(total) can include anycombination of power inputs. For example, the total system powerP_(total) can include the power consumption of various other fans withinrooftop unit 1402, the power consumption of a fluid pump, and/or anyother power consumption that occurs within vapor compression system1400.

Extremum seeking controller 502 is shown providing a temperaturesetpoint T_(sp) to a feedback controller 1404. In some embodiments, thetemperature setpoint T_(sp) is the manipulated variable which extremumseeking controller 502 adjusts to affect the total system powerP_(total). The temperature setpoint T_(sp) is a setpoint for thetemperature of the supply air 1420 leaving evaporator 1414. The supplyair temperature T_(sa) can be measured by temperature sensor 1418located downstream of evaporator 1414. Feedback controller 1404 is shownreceiving the supply air temperature T_(sa) as a feedback signal.

Feedback controller 1404 can operate evaporator fan 1416, condenser fan1422, and/or compressor 1406 to achieve the temperature setpoint T_(sp)provided by extremum seeking controller 502. For example, feedbackcontroller 1404 can increase the speed of evaporator fan 1416 toincrease the amount of heat removed from the supply air 1420 inevaporator 1414 or decrease the speed of evaporator fan 1416 to decreasethe amount of heat removed from the supply air 1420 in evaporator 1414.Similarly, feedback controller 1404 can increase the speed of condenserfan 1422 to increase the amount of heat removed from the refrigerant incondenser 1412 or decrease the speed of condenser fan 1422 to decreasethe amount of heat removed from the refrigerant in condenser 1412.

Extremum seeking controller 502 implements an extremum seeking controlstrategy that dynamically searches for an unknown input (e.g., optimalsupply air temperature setpoint T_(sp)) to obtain system performance(e.g., total power consumption P_(total)) that trends near optimal.Although feedback controller 1404 and extremum seeking controller 502are shown as separate devices, it is contemplated that feedbackcontroller 1404 and extremum seeking controller 502 can be combined intoa single device in some embodiments (e.g., a single controller thatperforms the functions of both extremum seeking controller 502 andfeedback controller 1404). For example, extremum seeking controller 502can be configured to control evaporator fan 1416, condenser fan 1422,and/or compressor 1406 directly without requiring an intermediatefeedback controller 1404.

Referring now to FIGS. 14B and 14C, a pair of flow diagrams 1450 and1470 illustrating the operation of extremum controller 502 in vaporcompression system 1400 are shown, according to some embodiments. Inboth flow diagrams 1450 and 1470, extremum seeking controller 502provides a temperature setpoint T_(sp) to a feedback controller 1404that operates to control supply air temperature T_(sa) in a vaporcompression system 1400 (blocks 1452 and 1472). Extremum seekingcontroller 502 can receive a total power consumption P_(total) of thevapor compression system 1400 as a feedback signal (blocks 1454 and1474).

In flow diagram 1450, extremum seeking controller 502 estimates agradient of the total power consumption P_(total) with respect to thesupply air temperature setpoint T_(sp) (block 1456). Extremum seekingcontroller 502 can provide control over the vapor compression system1400 by driving the obtained gradient toward zero by modulating thetemperature setpoint T_(sp) (block 1458). In some embodiments, extremumseeking controller 502 generates a stochastic excitation signal (block1460) and uses the stochastic excitation signal to generate a new supplyair temperature setpoint T_(sp). For example, extremum seekingcontroller 502 can generate the new temperature setpoint T_(sp) byperturbing the supply air temperature setpoint T_(sp) with thestochastic excitation signal (block 1462).

In flow diagram 1470, extremum seeking controller 502 estimates anormalized correlation coefficient relating the total power consumptionP total to the supply air temperature setpoint T_(sp) (block 1476).Extremum seeking controller 502 can provide control over the vaporcompression system 1400 by driving the estimated correlation coefficienttoward zero by modulating the temperature setpoint T_(sp) (block 1478).In some embodiments, extremum seeking controller 502 generates anexcitation signal (block 1480) and uses the excitation signal togenerate a new supply air temperature setpoint T_(sp). For example,extremum seeking controller 502 can generate the new temperaturesetpoint T_(sp) by perturbing the supply air temperature setpoint T_(sp)with the excitation signal (block 1482).

Vapor Compression System 1500

Referring now to FIG. 15A, another vapor compression air conditioningsystem 1500 is shown, according to some embodiments. System 1500 caninclude some or all of the components of vapor compression system 1400,as described with reference to FIG. 14A. For example, system 1500 isshown to include a refrigerant circuit 1510. Refrigerant circuit 1510includes a condenser 1512, an evaporator 1514, an expansion valve 1524,and a compressor 1506. Compressor 1506 is configured to circulate arefrigerant between evaporator 1514 and condenser 1512. Refrigerantcircuit 1510 operates using a vapor compression cycle. For example,compressor 1506 compresses the refrigerant to a hot, high pressurestate. The compressed refrigerant flows through condenser 1512 where therefrigerant rejects heat. A condenser fan 1522 can be used to modulatethe rate of heat transfer within condenser 1512. The cooled refrigerantis expanded by expansion valve 1524 to a low pressure, low temperaturestate. The expanded refrigerant flows through evaporator 1514 where therefrigerant absorbs heat. An evaporator fan 1516 can be used to modulatethe rate of heat transfer within evaporator 1514.

In some embodiments, refrigerant circuit 1510 is located within arooftop unit 1502 (e.g., a rooftop air handling unit) as shown in FIG.15A. Rooftop unit 1502 can be configured to provide cooling for supplyair 1520 flowing through an air duct 1522. For example, evaporator 1514can be located within air duct 1522 such that supply air 1520 flowsthrough evaporator 1514 and is cooled by transferring heat to theexpanded refrigerant within evaporator 1514. The cooled airflow can thenbe routed to a building to provide cooling for a room or zone of thebuilding. The temperature of supply air 1520 can be measured by atemperature sensor 1518 located downstream of evaporator 1514 (e.g.,within duct 1522). In other embodiments, refrigerant circuit 1510 can beused in any of a variety of other systems or devices that transfer heatusing a vapor compression cycle (e.g., chillers, heat pumps, heatrecovery chillers, refrigeration devices, etc.).

Extremum seeking controller 502 is shown receiving a power inputP_(total) representing the total power consumed by compressor 1506P_(comp), evaporator fan 1516 P_(fan,evap), and condenser fan 1522P_(fan,cond) (i.e., P_(total)=P_(comp)+P_(fan,evap)+P_(fan,cond)). Asillustrated in FIG. 15A, the power inputs P_(comp) P_(fan,evap), andP_(fan,cond) can be summed outside of extremum seeking controller 502 atsumming block 1508 to provide a combined signal representative of thetotal power P_(total). In other embodiments, extremum seeking controller502 receives the individual power inputs P_(comp), P_(fan,evap), andP_(fan,cond) and conducts the summation of summing block 1508. In eithercase, extremum seeking controller 502 can be said to receive the powerinputs P_(comp), P_(fan,evap), and P_(fan,cond) even if the power inputsare provided as a single summed or combined signal P_(total)representing the total system power.

In some embodiments, the total system power P_(total) is the performancevariable which extremum seeking controller 502 seeks to optimize (e.g.,minimize). The total system power P_(total) can include the powerconsumption of one or more components of vapor compression system 1500.In the embodiment shown in FIG. 15A, the total system power P_(total)includes P_(comp), P_(fan,evap), and P_(fan,cond). However, in variousother embodiments, the total system power P_(total) can include anycombination of power inputs. For example, the total system powerP_(total) can include the power consumption of various other fans withinrooftop unit 1502, the power consumption of a fluid pump, and/or anyother power consumption that occurs within vapor compression system1500.

Extremum seeking controller 502 is shown providing a control signalregulating the fan speed S_(sp) to evaporator fan 1516. In someembodiments, the fan speed S_(sp) is the manipulated variable whichextremum seeking controller 502 adjusts to affect the total system powerP_(total). Increasing the fan speed S_(sp) can increase the amount ofheat removed from the supply air 1520 in evaporator 1514 and increasethe total system power consumption P_(total). Similarly, decreasing thefan speed S_(sp) can decrease the amount of heat removed from the supplyair 1520 in evaporator 1514 and decrease the total system powerconsumption P_(total.) Extremum seeking controller 502 implements anextremum seeking control strategy that dynamically searches for anunknown input (e.g., optimal evaporator fan speed S_(sp)) to obtainsystem performance (e.g., total power consumption P_(total)) that trendsnear optimal.

Referring now to FIGS. 15B and 15C, a pair of flow diagrams 1550 and1570 illustrating the operation of extremum controller 502 in vaporcompression system 1500 are shown, according to some embodiments. Inboth flow diagrams 1550 and 1570, extremum seeking controller 502provides a control signal regulating a fan speed S_(sp) to an evaporatorfan 1516 in a vapor compression system 1500 (blocks 1552 and 1572).Extremum seeking controller 502 can receive a total power consumptionP_(total) of the vapor compression system 1500 as a feedback signal(blocks 1554 and 1574).

In flow diagram 1550, extremum seeking controller 502 estimates agradient of the total power consumption P_(total) with respect to theevaporator fan speed S_(sp) (block 1556). Extremum seeking controller502 can provide control over the vapor compression system 1500 bydriving the obtained gradient toward zero by modulating the evaporatorfan speed S_(sp) (block 1558). In some embodiments, extremum seekingcontroller 502 generates a stochastic excitation signal (block 1560) anduses the stochastic excitation signal to generate a new evaporator fanspeed S_(sp). For example, extremum seeking controller 502 can generatethe new evaporator fan speed S_(sp) by perturbing the evaporator fanspeed S_(sp) with the stochastic excitation signal (block 1562).

In flow diagram 1570, extremum seeking controller 502 estimates anormalized correlation coefficient relating the total power consumptionP total to the evaporator fan speed S_(sp) (block 1576). Extremumseeking controller 502 can provide control over the vapor compressionsystem 1500 by driving the estimated correlation coefficient toward zeroby modulating the evaporator fan speed S_(sp) (block 1578). In someembodiments, extremum seeking controller 502 generates an excitationsignal (block 1580) and uses the excitation signal to generate a newcontrol signal for the evaporator fan. For example, extremum seekingcontroller 502 can generate the new speed control signal by perturbingthe evaporator fan speed S_(sp) with the excitation signal (block 1582).

Vapor Compression System 1600

Referring now to FIG. 16A, a vapor compression air conditioning system1600 is shown, according to some embodiments. System 1600 is shown toinclude a refrigerant circuit 1610. Refrigerant circuit 1610 includes acondenser 1612, an evaporator 1614, an expansion valve 1624, and acompressor 1606. Compressor 1606 is configured to circulate arefrigerant between evaporator 1614 and condenser 1612. Refrigerantcircuit 1610 operates using a vapor compression cycle. For example,compressor 1606 compresses the refrigerant to a hot, high pressurestate. The compressed refrigerant flows through condenser 1612 where therefrigerant rejects heat. A condenser fan 1622 can be used to modulatethe rate of heat transfer within condenser 1612. The cooled refrigerantis expanded by expansion valve 1624 to a low pressure, low temperaturestate. The expanded refrigerant flows through evaporator 1614 where therefrigerant absorbs heat. An evaporator fan 1616 can be used to modulatethe rate of heat transfer within evaporator 1614.

In some embodiments, refrigerant circuit 1610 is located within arooftop unit 1602 (e.g., a rooftop air handling unit) as shown in FIG.16A. Rooftop unit 1602 can be configured to provide cooling for supplyair 1620 flowing through an air duct 1622. For example, evaporator 1614can be located within air duct 1622 such that supply air 1620 flowsthrough evaporator 1614 and is cooled by transferring heat to theexpanded refrigerant within evaporator 1614. The cooled airflow can thenbe routed to a building to provide cooling for a room or zone of thebuilding. The temperature of supply air 1620 can be measured by atemperature sensor 1618 located downstream of evaporator 1614 (e.g.,within duct 1622). In other embodiments, refrigerant circuit 1610 can beused in any of a variety of other systems or devices that transfer heatusing a vapor compression cycle (e.g., chillers, heat pumps, heatrecovery chillers, refrigeration devices, etc.).

Extremum seeking controller 502 is shown receiving a power inputP_(total) representing the total power consumed by compressor 1606P_(comp), evaporator fan 1616 P_(fan,evap), and condenser fan 1622P_(fan,cond) (i.e., P_(total)=P_(comp)+P_(fan,evap)+P_(fan,cond)). Asillustrated in FIG. 16A, the power inputs P_(comp), P_(fan,evap), andP_(fan,cond) can be summed outside of extremum seeking controller 502 atsumming block 1608 to provide a combined signal representative of thetotal power P_(total). In other embodiments, extremum seeking controller502 receives the individual power inputs P_(comp), P_(fan,evap), andP_(fan,cond) and conducts the summation of summing block 1608. In eithercase, extremum seeking controller 502 can be said to receive the powerinputs P_(camp), P_(fan,evap), and P_(fan,cond) even if the power inputsare provided as a single summed or combined signal P_(total)representing the total system power.

In some embodiments, the total system power P_(total) is the performancevariable which extremum seeking controller 502 seeks to optimize (e.g.,minimize). The total system power P_(total) can include the powerconsumption of one or more components of vapor compression system 1600.In the embodiment shown in FIG. 16A, the total system power P_(total)includes P_(comp), P_(fan,evap), and P_(fan,cond). However, in variousother embodiments, the total system power P_(total) can include anycombination of power inputs. For example, the total system powerP_(total) can include the power consumption of various other fans withinrooftop unit 1602, the power consumption of a fluid pump, and/or anyother power consumption that occurs within vapor compression system1600.

Extremum seeking controller 502 is shown providing a temperaturesetpoint T_(sp) to a feedback controller 1604 and a control signalregulating a fan speed S_(sp) to condenser fan 1622. In someembodiments, the temperature setpoint T_(sp) and the condenser fan speedS_(sp) are the manipulated variables which extremum seeking controller502 adjusts to affect the total system power P_(total). The temperaturesetpoint T_(sp) is a setpoint for the temperature of the supply air 1620leaving evaporator 1614. The supply air temperature T_(sa) can bemeasured by temperature sensor 1618 located downstream of evaporator1614. Feedback controller 1604 is shown receiving the supply airtemperature T_(sa) as a feedback signal. The fan speed S_(sp) is thespeed of condenser fan 1622.

Feedback controller 1604 can operate evaporator fan 1616 and/orcompressor 1606 to achieve the temperature setpoint T_(sp) provided byextremum seeking controller 502. For example, feedback controller 1604can increase the speed of evaporator fan 1616 to increase the amount ofheat removed from the supply air 1620 in evaporator 1614 or decrease thespeed of evaporator fan 1616 to decrease the amount of heat removed fromthe supply air 1620 in evaporator 1614. Similarly, extremum seekingcontroller 502 can modulate the condenser fan speed S_(sa) to increasethe amount of heat removed from the refrigerant in condenser 1612 (e.g.,by increasing the condenser fan speed S_(sa)) or decrease the amount ofheat removed from the refrigerant in condenser 1612 (e.g., by decreasingthe condenser fan speed S_(sa)).

Extremum seeking controller 502 implements an extremum seeking controlstrategy that dynamically searches for unknown inputs (e.g., optimalsupply air temperature setpoint T_(sp) and/or optimal condenser fanspeed S_(sa)) to obtain system performance (e.g., total powerconsumption P_(total)) that trends near optimal. Although feedbackcontroller 1604 and extremum seeking controller 502 are shown asseparate devices, it is contemplated that feedback controller 1604 andextremum seeking controller 502 can be combined into a single device insome embodiments (e.g., a single controller that performs the functionsof both extremum seeking controller 502 and feedback controller 1604).For example, extremum seeking controller 502 can be configured tocontrol evaporator fan 1616, condenser fan 1622, and/or compressor 1606directly without requiring an intermediate feedback controller 1604.

Referring now to FIGS. 16B and 16C, a pair of flow diagrams 1650 and1670 illustrating the operation of extremum controller 502 in vaporcompression system 1600 are shown, according to some embodiments. Inboth flow diagrams 1650 and 1670, extremum seeking controller 502provides a temperature setpoint T_(sp) to a feedback controller 1604that operates to control supply air temperature T_(sa) in a vaporcompression system 1600 (blocks 1652 and 1672). Extremum seekingcontroller 502 also provides a control signal that regulates a fan speedto a condenser fan 1622 in the vapor compression system 1600 (blocks1653 and 1674). Extremum seeking controller 502 can receive a totalpower consumption P_(total) of the vapor compression system 1600 as afeedback signal (blocks 1654 and 1674).

In flow diagram 1650, extremum seeking controller 502 estimates a firstgradient of the total power consumption P_(total) with respect to thesupply air temperature setpoint T_(sp) and a second gradient of thetotal power consumption P_(total) with respect to the condenser fanspeed S_(sp) (block 1656). Extremum seeking controller 502 can providecontrol over the vapor compression system 1600 by driving the obtainedgradients toward zero by modulating the temperature setpoint T_(sp)and/or the condenser fan speed S_(sp) (block 1658). In some embodiments,extremum seeking controller 502 generates stochastic excitation signals(block 1660) and uses the stochastic excitation signals to generate anew supply air temperature setpoint T_(sp) and a new control signalregulating the condenser fan speed S_(sp). For example, extremum seekingcontroller 502 can generate the new temperature setpoint T_(sp) byperturbing the supply air temperature setpoint T_(sp) with a firststochastic excitation signal and can generate the new control signal forthe condenser fan 1622 by perturbing the condenser fan speed S_(sp) witha second stochastic excitation signal (block 1662).

In flow diagram 1670, extremum seeking controller 502 estimates a firstnormalized correlation coefficient relating the total power consumptionP total to the supply air temperature setpoint T_(sp) and a secondnormalized correlation coefficient relating the total power consumptionP_(total) to the condenser fan speed S_(sp) (block 1676). Extremumseeking controller 502 can provide control over the vapor compressionsystem 1600 by driving the estimated correlation coefficients towardzero by modulating the temperature setpoint T_(sp) and/or the condenserfan speed S_(sp) (block 1678). In some embodiments, extremum seekingcontroller 502 generates excitation signals (block 1680) and uses theexcitation signal to generate a new supply air temperature setpointT_(sp) and a new control signal regulating the condenser fan speedS_(sp). For example, extremum seeking controller 502 can generate thenew temperature setpoint T_(sp) by perturbing the supply air temperaturesetpoint T_(sp) with a first excitation signal and can generate the newcontrol signal for the condenser fan 1622 by perturbing the condenserfan speed S_(sp) with a second excitation signal (block 1682).

Configuration of Exemplary Embodiments

The construction and arrangement of the systems and methods as shown inthe various exemplary embodiments are illustrative only. Although only afew embodiments have been described in detail in this disclosure, manymodifications are possible (e.g., variations in sizes, dimensions,structures, shapes and proportions of the various elements, values ofparameters, mounting arrangements, use of materials, colors,orientations, etc.). For example, the position of elements can bereversed or otherwise varied and the nature or number of discreteelements or positions can be altered or varied. Accordingly, all suchmodifications are intended to be included within the scope of thepresent disclosure. The order or sequence of any process or method stepscan be varied or re-sequenced according to alternative embodiments.Other substitutions, modifications, changes, and omissions can be madein the design, operating conditions and arrangement of the exemplaryembodiments without departing from the scope of the present disclosure.

The present disclosure contemplates methods, systems and programproducts on any machine-readable media for accomplishing variousoperations. The embodiments of the present disclosure can be implementedusing existing computer processors, or by a special purpose computerprocessor for an appropriate system, incorporated for this or anotherpurpose, or by a hardwired system. Embodiments within the scope of thepresent disclosure include program products comprising machine-readablemedia for carrying or having machine-executable instructions or datastructures stored thereon. Such machine-readable media can be anyavailable media that can be accessed by a general purpose or specialpurpose computer or other machine with a processor. By way of example,such machine-readable media can comprise RAM, ROM, EPROM, EEPROM, CD-ROMor other optical disk storage, magnetic disk storage or other magneticstorage devices, or any other medium which can be used to carry or storedesired program code in the form of machine-executable instructions ordata structures and which can be accessed by a general purpose orspecial purpose computer or other machine with a processor. Combinationsof the above are also included within the scope of machine-readablemedia. Machine-executable instructions include, for example,instructions and data which cause a general purpose computer, specialpurpose computer, or special purpose processing machines to perform acertain function or group of functions.

Although the figures show a specific order of method steps, the order ofthe steps may differ from what is depicted. Also two or more steps canbe performed concurrently or with partial concurrence. Such variationwill depend on the software and hardware systems chosen and on designerchoice. All such variations are within the scope of the disclosure.Likewise, software implementations could be accomplished with standardprogramming techniques with rule based logic and other logic toaccomplish the various connection steps, processing steps, comparisonsteps and decision steps.

What is claimed is:
 1. A control system for a plant, the control systemcomprising: a feedback controller for operating the plant to achieve avalue of a manipulated variable; and an extremum-seeking controllerconfigured to provide the value of the manipulated variable to thefeedback controller and to determine a value for the manipulatedvariable by: perturbing the manipulated variable with an excitationsignal; monitoring a performance variable of the plant resulting fromthe perturbed manipulated variable; estimating a normalized correlationcoefficient relating the performance variable to the manipulatedvariable; modulating the manipulated variable to drive the normalizedcorrelation coefficient toward zero using a general set of tuningparameters, wherein the general set of tuning parameters are adapted foruse with the normalized correlation coefficient, independent of a scaleof the performance variable.
 2. The control system of claim 1, whereinthe excitation signal is a non periodic signal comprising at least oneof a random walk signal, a non deterministic signal, and a non-repeatingsignal.
 3. The control system of claim 1, wherein the value of themanipulated variable comprises: a stochastic portion defined by astochastic excitation signal; and a non-stochastic portion determined bydriving the normalized correlation coefficient to zero.
 4. The controlsystem of claim 1, wherein the extremum-seeking controller comprises anintegrator configured to generate the excitation signal by integrating arandom noise signal.
 5. The control system of claim 1, wherein theextremum-seeking controller is configured to estimate the normalizedcorrelation coefficient relating the performance variable to themanipulated variable by performing a recursive least squares estimationprocess with exponential forgetting.
 6. The control system of claim 1,wherein the extremum-seeking controller is configured to estimate agradient of the performance variable with respect to the manipulatedvariable by performing a regression process.
 7. The control system ofclaim 6, wherein the regression process comprises: obtaining a linearmodel for the performance variable, the linear model defining theperformance variable as a linear function of the manipulated variable,an offset parameter, and a gradient parameter; estimating a value forthe gradient parameter based on an observed value for the performancevariable and an observed value for the manipulated variable; and usingthe estimated value for the gradient parameter as the gradient of theperformance variable with respect to the manipulated variable.
 8. Thecontrol system of claim 1, wherein the feedback controller is configuredto achieve the manipulated variable by adjusting operation of equipmentof the plant.
 9. An extremum-seeking controller for a plant comprising:one or more processors; and one or more non-transitory computer-readablemedia storing instructions that, when executed by the one or moreprocessors, cause the one or more processors to perform operationscomprising: perturbing a manipulated variable with an excitation signaland providing the perturbed manipulated variable as an input to a plant;monitoring a performance variable of the plant resulting from theperturbed manipulated variable; estimating a normalized correlationcoefficient relating the performance variable to the manipulatedvariable; and modulating the manipulated variable to drive thenormalized correlation coefficient toward zero using a general set oftuning parameters, wherein the general set of tuning parameters areadapted for use with the normalized correlation coefficient, independentof a scale of the performance variable.
 10. The extremum-seekingcontroller of claim 9, wherein the normalized correlation coefficient isestimated with a recursive estimation process, wherein the recursiveestimation process is a recursive least squares estimation process withexponential forgetting.
 11. The extremum-seeking controller of claim 9,wherein the one or more processors are configured to perform therecursive estimation process for the manipulated variable by:calculating a covariance between the performance variable and themanipulated variable; calculating a variance of the manipulatedvariable; and using the calculated covariance and the calculatedvariance to estimate the normalized correlation coefficient relating theperformance variable to the manipulated variable.
 12. Theextremum-seeking controller of claim 9, wherein the one or moreprocessors are configured to perform the recursive estimation processfor the manipulated variable by: calculating an exponentially-weightedmoving average (EWMA) of a plurality of samples of the manipulatedvariable; calculating an EWMA of a plurality of samples of theperformance variable; and using the EWMAs to estimate the normalizedcorrelation coefficient relating the performance variable to themanipulated variable.
 13. The extremum-seeking controller of claim 9,wherein the recursive estimation process is a regression process. 14.The extremum-seeking controller of claim 13, wherein the one or moreprocessors are configured to perform the regression process by:obtaining a linear model for the performance variable, the linear modeldefining the performance variable as a linear function of themanipulated variable and a gradient parameter for the manipulatedvariable; estimating a value for the gradient parameter based onobserved value for the manipulated variable and an observed value forthe performance variable; and using the estimated values for thegradient parameter as the normalized correlation coefficients relatingthe performance variable to the manipulated variable.
 15. Theextremum-seeking controller of claim 9, wherein the excitation signal isa non periodic signal comprising at least one of a random walk signal, anon deterministic signal, and a non-repeating signal.
 16. Anextremum-seeking controller for a plant, the extremum-seeking controllercomprising: one or more processors; and one or more non-transitorycomputer-readable media storing instructions that, when executed by theone or more processors, cause the one or more processors to performoperations comprising: perturbing each of a plurality of manipulatedvariables with a different excitation signal; monitoring a performancevariable of the plant resulting from each of the perturbed manipulatedvariables; estimating normalized correlation coefficients relating theperformance variable to each of the perturbed manipulated variables; andmodulating the manipulated variables to drive the estimated normalizedcorrelation coefficients toward zero using a general set of tuningparameters, wherein the general set of tuning parameters are adapted foruse with the normalized correlation coefficient, independent of a scaleof the performance variable.
 17. The extremum-seeking controller ofclaim 16, wherein the one or more processors are configured to estimatethe normalized correlation coefficient for each manipulated variable by:calculating a covariance between each of the manipulated variables andthe performance variable; calculating a variance of each of themanipulated variables; calculating a variance of the performancevariable; and using the calculated covariance and the calculatedvariances to estimate the normalized correlation coefficient.
 18. Theextremum-seeking controller of claim 16, wherein the one or moreprocessors are configured to estimate the normalized correlationcoefficient for each of the manipulated variables by: estimating agradient of the performance variable with respect to each of themanipulated variables; calculating a standard deviation of each of themanipulated variables; calculating a standard deviation of theperformance variable; and using the estimated gradient and thecalculated standard deviations to estimate the normalized correlationcoefficient.
 19. The extremum-seeking controller of claim 16, whereinthe one or more processors are configured to estimate the normalizedcorrelation coefficient for each of the manipulated variables by:calculating an exponentially-weighted moving average (EWMA) of aplurality of samples of each of the manipulated variables; calculatingan EWMA of a plurality of samples of the performance variable; and usingthe EWMAs to estimate the normalized correlation coefficient.
 20. Theextremum-seeking controller of claim 16, wherein the excitation signalis a non periodic signal comprising at least one of a random walksignal, a non deterministic signal, and a non-repeating signal.